diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzjybc" "b/data_all_eng_slimpj/shuffled/split2/finalzzjybc" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzjybc" @@ -0,0 +1,5 @@ +{"text":"\\section*{Acknowledgements}\n\nA.S.\\ gratefully acknowledges very stimulating discussions with \nD. Kutasov; S.D.\\ and M.H.\\ thank A.\\ Gomberoff and\nC.\\ Teitelboim. The work of S.D.\\ was supported \nby NSF grant PHY 93-15811, that of\nM.H.\\ by I.I.S.N.\\ (Belgium), and that of A.S.\\ by\nthe Israeli Academy of Sciences and the Minerva Foundation\n(Germany). M.H.\\ and A.S.\\ thank Carmen Nu\\~ nez \nand the Bariloche High Energy Group for hospitality \nat the Bariloche Institute, where this work began.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nTo incentivize and enable the sharing and reuse of ML models, organizations are building repositories of pre-trained ML models, also known as \\textit{model zoos} or \\textit{model hubs}; known examples include HuggingFace \\cite{huggingface}, Tensorflow Hub\\cite{tensorflow}, and PyTorch Hub \\cite{pytorch}. \nThe metadata of a ML model can contain all the necessary information -- from a model's inference capabilities (e.g., identified object classes) to the architecture, execution time, training dataset, hyper-parameter configuration, and evaluation performance. Metadata management are developed in the context of an ongoing effort to promote Trustworthy and Responsible AI\\footnote{\\url{https:\/\/partnershiponai.org\/paper\/responsible-publication-recommendations\/}}. \n\nMetadata management for model zoos can facilitate MLOps and open up new opportunities for new and unexplored use cases. For instance: \n\\begin{enumerate*}[label=\\roman*)]\n \\item retrieving models from large repositories with complex filtering conditions; \n \\item continuous integration of models in production;\n \\item (semi-)automatic model composition; and\n \\item advanced data management system.\n\\end{enumerate*}\nWe will further discuss use cases in \\cref{sec:usecase}.\n\nThe potential of model zoos is currently hindered by the lack of a structured, queryable metadata format. Current repositories include a wide range of information, in a form of a model card \\cite{mitchell2019model}, but such information is mostly for human consumption, making it hard for automatic extension or management. At the same time, the level of the detail remains coarse-grained: for instance, Amazon SageMaker, AzureML, MLflow \\cite{zaharia2018accelerating} do not require mandatory reporting of the related metadata, except for model name and version. Practitioners have to search on external websites for further metadata information such as the data instances, and they even have to evaluate the model at hand in order to assess its performance. \n\n\\cref{tab:query} lists some example queries that are practical for ML developers and users, and we will further discuss them in \\cref{sec:usecase}. These queries require more fine-grained model metadata that current model repositories such as HuggingFace do not support. This highlights the need for detailed metadata of trained ML models and datasets in a structured and queryable representation.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n\n\n\nAs a step toward this goal, we present a metadata model to represent the metadata of ML models and related datasets. Based on this metadata model, we are currently developing a ML model metadata management platform (prototype available at \\href{http:\/\/modelsearch.io}{modelsearch.io}) that can be used to query such metadata. \n\nThe contributions of this paper go as follows:\n\n\n\n\\vspace{-4mm}\n\\begin{itemize}\n\\setlength\\itemsep{0em}\n \\item We showcase two use cases abstracted from ML life-cycle, which advocates the need to define appropriate metadata representation for model zoos (\\cref{sec:usecase}). \n \\item We propose a metadata model as the structured, queryable, and comprehensive metadata representation for model zoos (\\cref{ssec:model}).\n \\item We develop an early-stage tool to retrieve ML models and datasets based on our proposed metadata representation, which facilitates complex inference queries over model zoos (\\cref{ssec:tool}).\n \\end{itemize}\n \n\\vspace{-3mm}\n\\def1.2{1.2}\n\\begin{table}[t]\n \\centering\n \\caption{Example queries}\n \\label{tab:query}\n \\vskip 0.15in\n \\small\n \\begin{tabular}{p{0.3\\linewidth}|p{0.03\\linewidth}p{0.55\\linewidth}}\n \\toprule\n Property & ID & Query \\\\\n \n \\hline\n \\multirow{6}{*}{Dataset information} & \\multirow{3}{*}{1} & Retrieve text classification models trained on dataset crowdsourced by at least a group of 50 people \\\\ \\cline{2-3}\n & \\multirow{3}{*}{2} & Find a dataset collected from COCO and OpenImage with all the images containing ``dog'' \\\\\n \\hline\n \\multirow{5}{*}{Model performance} & \\multirow{3}{*}{3} & Retrieve models trained on ImageNet with an accuracy higher than 90\\% \\\\\n \\cline{2-3}\n & \\multirow{2}{*}{4} & Which model performs the best on COCO for person detection task? \\\\\n \\hline\n \n \n \\multirow{4}{*}{Interpretability} & \\multirow{2}{*}{5} & Retrieve a person detection model with no gender bias \\\\\n \\cline{2-3}\n & \\multirow{2}{*}{6} & Retrieve text generation models that do not generate hate speech \\\\\n \\hline\n \n \\multirow{3}{*}{Hardware-related} & \\multirow{3}{*}{7} & Retrieve image classification models that are suitable to deploy on edge devices\\\\\n \n \\hline\n \\end{tabular}\n\\vskip -0.1in\n\\end{table}\n\n\\section{Use Cases}\n\\label{sec:usecase}\n\n\n\nTo demonstrate the usefulness of a formalized metadata representation, in this section we explain two use cases with different stakeholders, where a comprehensive and well-structured metadata representation is needed.\n\n\n\n\\subsection{Metadata Retrieval Throughout the ML Life-cycle}\n\nThroughout the ML life-cycle, ML practitioners will require different metadata for tracking the ML model status, editing, comparing, or reporting.\nAn ML practitioner often needs to query models in large repositories with complex filtering conditions, e.g., data instance, performance, and inherit mechanism. \nIn \\cref{tab:query}, we list a few example queries revealing different properties of the metadata. For example, Query 1 and 2 require the metadata regarding the dataset, i.e., its attribute and source. Queries such as Query 3 to 6 require some other metadata properties. \nQuery 7, on the other hand, requires more complex information regarding the inference performance with specified hardware settings. For example, an edge device may have constraints such as limited computation power and storage. To answer this query, practitioners will need to obtain the model performance of different objectives, e.g., inference speed and memory footprint.\n\n\n\n\\subsection{(Semi-)Automatic Model Composition}\nNow we introduce a more advanced yet common use case.\nWith metadata being captured and well-represented, ML practitioners can make good use of the models trained offline and apply them to answer complex, ad-hoc inference queries. \n\nAs shown in \\cref{fig:optimizer}, the ad-hoc query can consist of multiple ML inference tasks with different dependencies and relations. Moreover, the query can be composed of specified constraints\/requirements (e.g., latency and accuracy restrictions).\nPractitioners can select a composition of models from the model zoo to answer the ad-hoc queries.\nFor example, an ML practitioner would like to design an application that can capture tweets with positive sentiment and a part-of-speech (PoS) tagger for data analysis. Since the data volume is significant and latency is also an essential factor to consider, the practitioner should select text classification model and PoS tagging model with fast inference speed. And the inference speed is greatly affected by the hardware being applied. If the application is deployed on the mobile phone, then the memory footprint is also a fundamental objective to be concerned with. \nTo identify which set of models could best address the query and constraints, they may require information regarding the model performance with different objectives (e.g., accuracy, inference speed, memory footprint). The metadata of the dataset can also provide information to detect concept drift, for instance.\n\n\\vspace{5px}\nState-of-the-art model repositories would not be able to support these use cases, as they often maintain limited metadata, like the model name or the training dataset. A first step towards accessible repository is the specification of a rich and structured metadata format. \n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.5\\textwidth]{figures-ml-inference-optmization.pdf}\n \\caption{ML inference query optimization}\n \\label{fig:optimizer}\n\n\\end{figure}\n\n\n\\section{Methodology}\n\nThis section defines a metadata model to represent different entities and relations. We also propose a practical tool to extract, store, and query the metadata. \nWith our proposed comprehensive metadata in a structured representation and the tool, users can address the use cases discussed in the previous section by retrieving metadata in fine-grained details.\n\n\n\n\\subsection{Metadata Model}\n\\label{ssec:model}\n\\cref{fig:model} presents a conceptual view of our proposed metadata model (from now on, \\emph{meta-model}), described using UML class diagrams. The \\emph{meta-model} composes of four packages: \n\\begin{enumerate*}[label=\\roman*)]\n \\item the \\texttt{Configuration} package, which defines the ML models associated with architecture, hyperparameter settings, input and output;\n \\item the \\texttt{Dataset} package, which represents information on datasets;\n \\item the \\texttt{Execution} package, which describes the inference results of the model, possibly enriched with description from a knowledge graph;\n \\item the \\texttt{Evaluation} package, which presents the run-time metrics obtained in specified hardware settings.\n\\end{enumerate*}\n\nDue to space limitation, and for the sake of clarity, we present a compact version of the \\emph{meta-model}, which is amenable to implementation in different formats, including relational databases and Linked Data. \n\n\n\\subsubsection{Configuration package}\n\nThe package composes of three classes: \\texttt{ML model}, \\texttt{Hyperparameter}, and \\texttt{Architecture}.\nThe \\texttt{ML Model} class encodes information about the \\textit{input} and \\textit{output} of the model, the prediction \\textit{task}, and the data \\textit{transformation} processes steps (e.g. data transformation and post-processing). For traditional ML model, the transformation pipelines can include steps for feature engineering.\nSuch a recording of the ML model details enables exploration tasks such as model selection, data lineage, and visualization. It also helps reproduce the ML model and compare it to others from different perspectives. The \\texttt{Hyperparameter}, as the name suggests, capture the type and values of hyper-parameters used to train the specific model. Finally, the \\texttt{Architecture} class contains information about the specific structural characteristic of a model, e.g. if it is a Feed-Forward Neural Network. \n\n\n\\subsubsection{Dataset Package}\n\nThe behaviour of a ML model heavily rely on the data it used. \nThus the \\emph{meta-model} includes a \\texttt{Dataset} packaged, representing both \\texttt{Dataset}s and their \\textit{Data Instance}s.\nWith the \\texttt{Dataset} element, we present the metadata of the datasets that is significant for data management and reporting, e.g., data source, data version. \nWith \\texttt{Data Instance} we facilitate data provenance, for which it is of utmost importance to know which instances the model is trained on and how the data is utilized. \nMoreover, it is also significant to know if the data has revealed sensitive information, for example, violating the GDPR regulations. \n\n\n\n\n\n\\begin{figure}[t]\n\\begin{center}\n \\includegraphics[width=0.51\\textwidth]{uml-class.pdf}\n \\caption{Modeling the metadata throughout ML life-cycle}\n \\label{fig:model}\n\\end{center}\n \\vskip -0.2in\n\\end{figure}\n\n\\subsubsection{Execution Package}\nThe \\texttt{Execution} package includes two important classes. The \\texttt{Model Prediction} class allows the description of inference metrics on one or more prediction output. Optionally a \\texttt{Model Prediction} can be linked with a \\emph{Semantic Concept} from a knowledge base, thus allowing complex reasoning. Consider the example that an ML practitioner is building an ML model for image classification of cars, and she tries to conduct model diagnosis. She may have several questions: what makes the model identify a car as a car? What are the semantic concepts that the model is capable to identify? Are the wheels that make it believe that it is a car? To support such use cases, the \\emph{meta-model} allows to store information about inference performance on specific data instances, which can be used to describe the the behavior of a model, i.e., in what circumstances a model can perform well and why. Such information and awareness of the model prediction, may significantly improve ML model interpretability in various applications such as health care, law enforcement and finance.\n\n\n\\subsubsection{Evaluation Package}\n\nThe performance evaluation of an ML model is critical during both the training and deployment phases. The practitioners need to deploy a suitable ML model for the task given a specified environment, e.g., \\texttt{Hardware} parameters. A mismatch of the deployment will lead to latency issues or reliability concerns, which results in user dissatisfaction. Thus a well-rounded evaluation of the model performance is significant. Hence, we have the \\texttt{Hardware} preserving the hardware settings and \\texttt{Evaluation Metric} with metric-related metadata. Different models will need different evaluation metrics, for instance, a regression model would have a \\emph{Mean Square Error} - \\textit{MSE}, while a multi-class classification model would have an \\emph{accuracy} value for each class.\n\n\n\\begin{figure}[t]\n \\centering\n \\includegraphics[width=0.4\\textwidth]{structure.pdf}\n \\caption{The structure of our metadata management tool}\n \\label{fig:structure}\n \n\\end{figure}\n\n\\subsection{A Metadata Management Tool}\n\\label{ssec:tool}\nWe are currently developing a tool\\footnote{\\url{https:\/\/modelsearch.io}} to manage the metadata for model zoos, whose structure is shown in \\cref{fig:structure}. \n\nWe collect metadata in the following three ways.\n\\begin{enumerate*}[label=(\\roman*)]\n \n \\item a user can add the metadata regarding a model or a dataset by filling in specified fields. Then such information is translated by the \\emph{Ingestion API} into structured representation according to the above-mentioned metadata model, and stored in the \\emph{Metadata Storage}. \n \n \\item The tool can also automatically extract information from external model zoos, e.g., HuggingFace and PyTorch Hub. We can extract metadata from their API or information on the web page and record the source of the external metadata in our metadata field. \n \n \\item To gather the information regarding the model performance, we apply a third way to obtain the metadata. We obtain the performance evaluation by executing the model on a dataset with specified hardware settings. This process can be done by utilizing cloud resources.\n\\end{enumerate*}\n \nBy the time we are writing the paper, the tool is available to crawl information from the external model zoos and support querying the metadata stored in structured and queryable representations. Besides ingesting, extracting, and storing the metadata, our proposed tool allows a user to (i) retrieve the models that help them identify a model, (ii) compare multiple models, (iii) or explore the properties of models\/data by composing queries on the metadata.\n\n\n\n\n\\section{Related work}\n\nRecent studies focus on different aspects of management during ML life-cycle, from model versioning, model reporting to model evaluation. Each is important for practitioners to manage and understand the models. We observe a gap among the profound works, a comprehensive and queryable metadata representation. With the metadata representation, we can thus better manage the ML models and data, including the interactions between them.\n\n\n\\emph{Metadata Captured in ML Versioning and Data Versioning}\n\nRecent works developed tools\/systems to manage ML models and datasets used in ML life-cycle. Works such as Modeldb \\cite{vartak2016modeldb}, ModelHub \\cite{miao2017towards} and MLflow \\cite{zaharia2018accelerating} developed tools to manage ML mdoels with different versions and captured the metadata of the models in different levels of details. However, they focused on the abstractions on the model, they lack information on the model performance under different hardware settings (e.g., inference speed, accuracy, memory footprint), and it paid little attention on the datasets that the models consume. \nFor ML dataset management and versioning, research work such as Mldp \\cite{agrawal2019data} and DataLab \\cite{zhang2016datalab} can be complements to the above-mentioned model management system by supporting data management. \n\n\n\\emph{Model Cards and Data Cards}\n\nRecent research also focus on the reporting of models and dataset, covering aspects not only limited to basic informative components, but also including ethical, inclusive and fair considerations. Model cards \\cite{mitchell2019model}, for example, proposed to include information regarding model intended use cases, potential pitfalls and other contexts that can improve model understanding. Similar idea also lies in data cards\/sheets. Examples include \\cite{gebru2021datasheets,miceli2021documenting,boyd2021datasheets}. Though model cards and data cards contain rich information, the Q\\&A format is nonetheless unfriendly to machine to process and thus cannot be easily managed and retrieved. \n\n\\emph{Model Performance Benchmarking}\n\nA growing body of published work also focus into the benchmarking of ML model performance, such as MLperf \\cite{mattson2020mlperf,reddi2020mlperf}, fathom \\cite{adolf2016fathom}, Dawnbench \\cite{coleman2017dawnbench}. These platforms covered a set of meatadata including metrics, training\/inference configurations with specified hardware\/software setting. Their focus is the report of the model performance at different ML life-cycle stages (training or inference). They paid little attention on the dataset the model used, whose path is provided as an argument filled by the user. The model process pipeline is also not covered besides the model scripts.\n\n\n\n\n\\section{Conclusion and outlook}\nIn this paper, we advocate for the need of a structured, queryable, and comprehensive metadata representation for model zoos. \nWe propose a metadata model for such metadata representation to tackle different use cases. We also develop a tool that helps practitioners to manage and query the metadata.\nFuture work can develop tools\/frameworks to extract more useful information from the textual description in the model\/data cards by applying natural language processing techniques and mapping it to the predefined metadata representation.\n\n\n\n\n\n\n\n\\newpage\n\\nocite{langley00}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\\label{section:introduction}\n\n\\noindent\nIn this work, we prove optimal convergence rates for an adaptive isogeometric boundary element method for the (first-kind) hyper-singular integral equation \n\\begin{align}\\label{eq:hyper strong}\n \\mathfrak{W}u=g:=(1\/2-\\mathfrak{K'})\\phi\n \\quad\\text{on }\\Gamma:=\\partial\\Omega\n\\end{align}\nassociated with the 2D Laplacian. Here, $\\Omega \\subset {\\mathbb R}^2$ is a bounded Lipschitz domain, whose boundary can be parametrized via non-uniform rational B-splines (NURBS); see Section~\\ref{section:preliminaries} for the precise statement of the integral operators $\\mathfrak{W}$ and $\\mathfrak{K}'$ as well as for definition and properties of NURBS. Given boundary data $\\phi$, we seek for the unknown integral density $u$. We note that~\\eqref{eq:hyper strong} is equivalent to the Laplace--Neumann problem\n\\begin{align}\n -\\Delta P = 0 \\text{ in } \\Omega\n \\quad \\text{subject to Neumann boundary conditions} \\quad\n \\partial P\/\\partial \\nu = \\phi \\text{ on } \\Gamma,\n\\end{align}\nwhere $u = P|_\\Gamma$ is the trace of the sought potential $P$.\n\nThe central idea of isogeometric analysis (IGA) is to use the same ansatz functions for the discretization of \\eqref{eq:hyper strong}, as are used for the representation of the problem geometry in CAD.\n This concept, originally invented in~\\cite{hughes2005} for finite element methods (IGAFEM) has proved very fruitful in applications; see also the monograph \\cite{bible}. \nSince CAD directly provides a parametrization of the boundary $\\partial \\Omega$, this makes the boundary element method (BEM) the most attractive numerical scheme, if applicable (i.e., provided that the fundamental solution of the differential operator is explicitly known); see \\cite{igabem2d,igabem3d} for the first works on isogeometric BEM (IGABEM) for 2D resp.\\ 3D. \n\nWe refer {to~\\cite{SBTR,helmholtziga,simpson,adss16,tran}} for numerical experiments, to \\cite{cad2wave,TM,zechner,dhp16,wolf18,wolf_new} for fast IGABEM based on wavelets, fast multipole, $\\mathcal{H}$-matrices resp.\\ $\\mathcal{H}^2$-matrices, and \nto~\\cite{stokesiga,keuchel,acdsss18,acdsss18,fgkss18} for some quadrature analysis.\n\n\n\n\nOn the one hand, IGA naturally leads to high-order ansatz functions. On the other hand, however, optimal convergence behavior with higher-order discretizations is only observed in simulations, if the (given) data $\\phi$ as well as the (unknown) solution $u$ are smooth. Therefore, {\\sl a~posteriori} error estimation and related adaptive strategies are mandatory to realize the full potential of IGA. Rate-optimal adaptive strategies for IGAFEM have been proposed and analyzed independently in~\\cite{bg2017,ghp2017} for IGAFEM, while the earlier work~\\cite{bg2016} proves only linear convergence. As far as IGABEM is concerned, available results focus on the weakly-singular integral equation with energy space $H^{-1\/2}(\\Gamma)$; see~\\cite{igafaermann,resigabem} for {\\sl a~posteriori} error estimation as well as~\\cite{optigabem} for the analysis of a rate-optimal adaptive IGABEM in 2D, and~\\cite{gantner17} for corresponding results for IGABEM in 3D with hierarchical splines. \nRecently, \\cite{priga} investigated optimal preconditioning for IGABEM in 2D with locally refined meshes.\n\nIn this work, we consider the hyper-singular integral equation~\\eqref{eq:hyper strong} with energy space $H^{1\/2}(\\Gamma)$. We stress that the latter is more challenging than the weakly-singular case, with respect to numerical analysis as well as stability of numerical simulations. Moreover, the present work addresses also the adaptive steering of the smoothness of the NURBS ansatz spaces across elements. The adaptive strategy thus goes beyond the classical \n\\begin{align*}\n \\boxed{\\rm\\, SOLVE\\,}\n \\,\\,\\longrightarrow\\,\\,\n \\boxed{\\rm\\, ESTIMATE\\,}\n \\,\\,\\longrightarrow\\,\\,\n \\boxed{\\rm\\, MARK\\,}\n \\,\\,\\longrightarrow\\,\\,\n \\boxed{\\rm\\, REFINE\\,}\n\\end{align*}\nconsidered, e.g., in~\\cite{fkmp,gantumur,part1,part2} for standard BEM with piecewise polynomials. Moreover, while the adaptive algorithm from~\\cite{optigabem} only allows for a smoothness reduction (which makes the ansatz space larger), the new algorithm also stears the local increase of smoothness (which makes the ansatz space smaller). \nAdditionally, we also account for the approximate computation of the right-hand side.\nWe prove that the new algorithm is rate optimal in the sense of~\\cite{axioms}. Moreover, as a side result, we observe that the related approximation classes are independent of the smoothness of the ansatz functions. \n\n\\defC_{\\rm eff}{C_{\\rm eff}}\nTo steer the algorithm, we adopt the weighted-residual error estimator from standard BEM~\\cite{cs95,carstensen97,cmps04,part2} and prove that it is reliable and weakly efficient,~i.e.,\n\\begin{subequations}\n\\begin{align}\n \\eta_\\bullet := \\big( \\norm{h_\\bullet^{1\/2} ( (1\/2-\\mathfrak{K}')\\phi_\\bullet - \\mathfrak{W}U_\\bullet )}{L^2(\\Gamma)}^2\n + \\norm{h_\\bullet^{1\/2} ( \\phi - \\phi_\\bullet )}{L^2(\\Gamma)}^2 \\big)^{1\/2}\n\\end{align}\nsatisfies (with the arclength derivative $\\partial_\\Gamma$) that \n\\begin{align}\n C_{\\rm rel}^{-1} \\, \\norm{u - U_\\bullet}{\\widetilde{H}^{1\/2}(\\Gamma)}\n \\le \\eta_\\bullet\n \\le C_{\\rm eff} \\big( \\norm{h_\\bullet^{1\/2} \\partial_\\Gamma( u - U_\\bullet )}{L^2(\\Gamma)}^2\n + \\norm{h_\\bullet^{1\/2} ( \\phi - \\phi_\\bullet )}{L^2(\\Gamma)}^2 \\big)^{1\/2}.\n\\end{align}\n\\end{subequations}\nHere, $h_\\bullet$ is the \\emph{local mesh-size}, and $U_\\bullet$ is the Galerkin solution with respect to some approximate discrete data $\\phi_\\bullet \\approx \\phi$. We compute $\\phi_\\bullet$ by the $L^2$-orthogonal projection of $\\phi$ onto discontinuous piecewise polynomials. We stress that data approximation is an important subject in numerical computations, and reliable numerical algorithms have to properly account for it. In particular, the benefit of our approach is that the implementation has to deal with discrete integral operators only. Since $\\phi$ is usually non-smooth with algebraic singularities, the stable numerical evaluation of $\\mathfrak{K}'\\phi$ would also require non-standard (and problem dependent) quadrature rules, which simultaneously resolve the logarithmic singularity of $\\mathfrak{K}'$ as well as the algebraic singularity of $\\phi$. \nThis is avoided by our approach. \nFinally, in the appendix, we generalize the presented results also to slit problems and the weakly-singular integral equation. \n\n\\subsection*{Outline}\nThe remainder of the work is organized as follows: \nSection~\\ref{section:preliminaries} provides the functional analytic setting of the boundary integral operators, the definition of the mesh, B-splines and NURBS together with their basic properties.\nIn Section~\\ref{sec:main result}, we introduce the new adaptive Algorithm~\\ref{the algorithm} and provide our main results on {\\sl a~posteriori} error analysis and optimal convergence in Theorem~\\ref{thm:main}.\nThe proof of the latter is postponed to Section~\\ref{sec:proof}, where we essentially verify the abstract \\textit{axioms of adaptivity} of \\cite{axioms} and sketch how they imply optimal convergence.\nAuxiliary results of general interest include a new Scott--Zhang-type operator onto rational splines (Section~\\ref{sec:scott zhang}) and inverse inequalities (Section~\\ref{sec:inverse inequalities}), which are well-known for standard BEM.\nIn Section~\\ref{section:numerics}, we underline our theoretical findings via numerical experiments. \nThere, we consider both the hyper-singular integral equation as well as weakly-singular integral equation.\nIndeed, the our results for the hyper-singular case are briefly generalized in the appendix, where we also comment on slit problems.\n\n\n\n\n\n\\section{Preliminaries}\n\\label{section:preliminaries}\n\\subsection{General notation}\nThroughout and without any ambiguity, $|\\cdot|$ denotes the absolute value of scalars, the Euclidean norm of vectors in ${\\mathbb R}^2$, the cardinality of a discrete set, the measure of a set in ${\\mathbb R}$ (e.g., the length of an interval), or the arclength of a curve in ${\\mathbb R}^{2}$.\nWe write $A\\lesssim B$ to abbreviate $A\\le cB$ with some generic constant $c>0$, which is clear from the context.\nMoreover, $A\\simeq B$ abbreviates $A\\lesssim B\\lesssim A$. \nThroughout, mesh-related quantities have the same index, e.g., ${\\mathcal N}_\\bullet$ is the set of nodes of the partition ${\\mathcal{Q}}_\\bullet$, and $h_\\bullet$ is the corresponding local mesh-width function etc. \nThe analogous notation is used for partitions ${\\mathcal{Q}}_\\circ$ resp.\\ ${\\mathcal{Q}}_\\ell$ etc. \nWe use \\,$\\widehat{\\cdot}$\\, to transform notation on the boundary to the parameter domain, e.g., $\\widehat{\\mathcal{Q}}_\\ell$ is the partition of the parameter domain corresponding to the partition ${\\mathcal{Q}}_\\ell$ of $\\Gamma$. \nThroughout, we make the the following convention: If ${\\mathcal N}_\\bullet$ is a set of nodes and $\\alpha_\\bullet(z)\\ge 0$ is defined for all $z\\in{\\mathcal N}_\\bullet$, then \n\\begin{align}\\label{eq:estimator convention}\n\\alpha_\\bullet:=\\alpha_\\bullet({\\mathcal N}_\\bullet), \\quad\\text{where}\\quad \\alpha_\\bullet(\\mathcal{S}_\\bullet)^2:=\\sum_{z\\in\\mathcal{S}_\\bullet}\\alpha_\\bullet(z)^2\\quad\\text{for all }\\mathcal{S}_\\bullet\\subseteq{\\mathcal N}_\\bullet.\n\\end{align}\n\n\n \n\n\n\\defC_\\Gamma{C_\\Gamma}\n\\subsection{Sobolev spaces}\n\\label{section:sobolev}\nThe usual Lebesgue and Sobolev spaces on $\\Gamma$ are denoted by $L^2(\\Gamma)=H^0(\\Gamma)$ and \n$H^1(\\Gamma)$. \nFor measurable $\\Gamma_0\\subseteq\\Gamma$, we define the corresponding seminorm \n \\begin{align}\n |u|_{H^{1}({\\Gamma_0})} := \\norm{\\partial_\\Gamma u}{L^2(\\Gamma_0)}\\quad \\text{for all }u\\in H^1(\\Gamma)\n \\end{align}\nwith the arclength derivative $\\partial_\\Gamma$.\nIt holds that \n\\begin{align}\n\\norm{u}{H^1(\\Gamma)}^2= \\norm{u}{L^2(\\Gamma)}^2+|u|_{H^1(\\Gamma)}^2\\quad\\text{for all }u\\in H^1(\\Gamma).\n\\end{align}\nMoreover, $\\widetilde{H}^1(\\Gamma)$ is the space of \n$H^1(\\Gamma)$ functions, which have a vanishing trace on the relative boundary \n$\\partial\\Gamma$ equipped with the same norm.\nSobolev spaces of fractional order $0<\\sigma<1$ are defined by\nthe\n$K$-method of interpolation~\\cite[Appendix B]{mclean}: \nFor $0<\\sigma<1$, let\n$H^\\sigma(\\Gamma) := [L^2(\\Gamma),H^1(\\Gamma)]_\\sigma$.\n\nFor $0<\\sigma\\le1$, Sobolev spaces of negative order are defined by duality\n$H^{\\mp\\sigma}(\\Gamma) := H^{\\pm\\sigma}(\\Gamma)^*$, \n where duality is understood with\nrespect to the extended $L^2(\\Gamma)$-scalar\nproduct $\\dual\\cdot\\cdot_\\Gamma$. \nFinally, we define $H^{\\pm \\sigma}_0(\\Gamma) = \\set{v\\in H^{\\pm \\sigma}(\\Gamma)}{\\dual{v}{1}_{\\Gamma} = 0}$\nfor all $0\\le \\sigma\\le1$.\n\nAll details and equivalent definitions of the Sobolev spaces \nare, for instance, found in the monographs~\\cite{hw,mclean,ss}.\n\n\n\\subsection{Hyper-singular integral equation}\n\\label{subsec:hypsing}%\nThe hyper-singular integral equation~\\eqref{eq:hyper strong} employs the hyper-singular operator $\\mathfrak{W}$ as well as the adjoint double-layer operator $\\mathfrak{K}'$.\nWith the fundamental solution $G(x,y):=-\\frac{1}{2\\pi}\\log|x-y|$ of the 2D Laplacian and the outer normal vector $\\nu$, \nthese have the following boundary integral representations\n\\begin{align}\n \\mathfrak{W}v(x)\n = \\frac{\\partial_x}{\\partial\\nu(x)}\\int_\\Gamma v(y)\\frac{\\partial_y}{\\partial\\nu(y)}G(x,y)\\,dy\n\\quad \\text{and} \\quad\n \\mathfrak{K}'\\psi(x)\n = \\int_\\Gamma \\psi(y)\\frac{\\partial_x}{\\partial\\nu(x)}G(x,y)\\,dy\n\\end{align}\nfor smooth densities $v,\\psi:\\Gamma\\to{\\mathbb R}$.\n\n\nFor $0\\le \\sigma\\le1$, the hyper-singular integral operator \n$\\mathfrak{W}:H^{\\sigma}(\\Gamma)\\to H^{\\sigma-1}(\\Gamma)$ and the adjoint double-layer operator $\\mathfrak{K}':H^{\\sigma-1}(\\Gamma)\\to H^\\sigma(\\Gamma)$ are well-defined, linear, and continuous. \n\n\nFor connected $\\Gamma=\\partial\\Omega$ and $\\sigma=1\/2$, the \noperator $\\mathfrak{W}$ is symmetric and elliptic up to the constant functions, i.e., \n$\\mathfrak{W}:H^{1\/2}_0(\\Gamma)\\to H^{-1\/2}_0(\\Gamma)$ is \nelliptic. In particular \n\\begin{align}\n\\edual{u}{v}:=\\dual{\\mathfrak{W} u}{v}_{\\Gamma} + \\dual{u}{1}_{\\Gamma}\\dual{v}{1}_{\\Gamma}\n\\end{align}\ndefines an equivalent scalar product on $H^{1\/2}(\\Gamma)$ with corresponding norm $\\enorm{\\cdot}$.\nMoreover, there holds the additional mapping property $\\mathfrak{K}':H_0^{-1\/2}(\\Gamma)\\to H_0^{-1\/2}(\\Gamma)$.\n\nWith this notation and provided that $\\phi\\in H^{-1\/2}_0(\\Gamma)$, the strong form~\\eqref{eq:hyper strong} is equivalently stated in variational form: Find $u\\in H^{1\/2}(\\Gamma)$ such that\n\\begin{align}\n\\label{eq:hyper weak}\n \\edual{u}{v} = \\dual{(1\/2-\\mathfrak{K}')\\phi}{v}_\\Gamma\n \\quad\\text{for all }v\\in H^{1\/2}(\\Gamma).\n\\end{align}\nTherefore, the Lax-Milgram lemma applies and proves that \\eqref{eq:hyper weak} resp.\\ \\eqref{eq:hyper strong} admits a unique solution $u\\in H^{1\/2}(\\Gamma)$.\nDetails are found, e.g., in \\cite{hw,mclean,ss,s}.\n\n\n\\subsection{Boundary parametrization}\n\\label{subsec:boundary parametrization}\nWe assume that $\\Gamma$ is parametrized by a continuous and piecewise continuously differentiable path $\\Gamma:[a,b]\\to \\Gamma$ such that $\\gamma|_{(a,b)}$ is injective.\nIn particular, $\\gamma|_{[a,b)}$ and $\\gamma|_{(a,b]}$ are bijective.\nThroughout and by abuse of notation, we write $\\gamma^{-1}$ for the inverse of $\\gamma|_{[a,b)}$ resp. $\\gamma|_{(a,b]}$.\nThe meaning will be clear from the context.\n\nFor the left- and right-hand derivative of $\\gamma$, we assume that {$\\gamma^{\\prime_\\ell}(t)\\neq 0$ for $t\\in(a,b]$ and $\\gamma^{\\prime_r}(t)\\neq 0$ for $t\\in [a,b)$.}\nMoreover, we assume for all $c>0$ that $\\gamma^{\\prime_\\ell}(t)\n+c\\gamma^{\\prime_r}(t)\\neq0$ for $t\\in(a,b)$ and $\\gamma^{\\prime_\\ell}(b)+c\\gamma^{\\prime_r}(a)\\neq0$.\n\n\n\\subsection{Boundary discretization}\n\\label{section:boundary:discrete}\nIn the following, we describe the different quantities, which define the discretization.\n\n{\\bf Nodes $\\boldsymbol{z_{\\bullet,j}=\\gamma(\\widehat{z}_{\\bullet,j})\\in\\mathcal{N}_\\bullet}$.}\\quad Let $\\mathcal{N}_\\bullet:=\\set{z_{\\bullet,j}}{j=1,\\dots,n_\\bullet}$ and $z_{\\bullet,0}:=z_{n_\\bullet}$ \n be a set of nodes. We suppose that $z_{\\bullet,j}=\\gamma(\\widehat{z}_{\\bullet,j})$ for some $\\widehat{z}_{\\bullet,j}\\in[a,b]$ with\n$a=\\widehat{z}_{\\bullet,0}<\\widehat{z}_{\\bullet,1}<\\widehat{z}_{\\bullet,2}<\\dots<\\widehat{z}_{\\bullet,n_\\bullet}=b$ such that \n$\\gamma|_{[\\widehat{z}_{\\bullet,j-1},\\widehat{z}_{\\bullet,j}]}\\in C^1([\\widehat{z}_{\\bullet,j-1},\\widehat{z}_{\\bullet,j}])$.\n\n{\\bf Multiplicity $\\boldsymbol{\\#_\\bullet z}$, $\\#_\\bullet \\mathcal{S}_\\bullet$, and knots $\\boldsymbol{{\\mathcal K}_\\bullet}$.}\\quad\nLet $p\\in{\\mathbb N}$ be some fixed polynomial order.\nEach interior node $z_{\\bullet,j}$ has a multiplicity $\\#_\\bullet z_{\\bullet,j}\\in\\{1,2\\dots, p\\}$ and $\\#_\\bullet {z}_{\\bullet,0}=\\#_\\bullet z_{n_\\bullet}=p+1$. \nFor $\\mathcal{S}_\\bullet\\subseteq{\\mathcal N}_\\bullet$, we set \n\\begin{align}\n\\#_\\bullet\\mathcal{S}_\\bullet:=\\sum_{z\\in\\mathcal{S}_\\bullet} \\#_\\bullet z.\n\\end{align}\nThe multiplicities induce the knot vector\n\\begin{align}\n{\\mathcal K}_\\bullet=(\\underbrace{z_{\\bullet,1},\\dots,z_{\\bullet,1}}_{\\#_\\bullet z_{\\bullet,1}-\\text{times}},\\dots,\\underbrace{z_{\\bullet,n_\\bullet},\\dots,z_{\\bullet,n_\\bullet}}_{\\#_\\bullet z_{\\bullet,n_\\bullet}-\\text{times}}).\n\\end{align} \n\n{\\bf Elements $\\boldsymbol{Q_{\\bullet,j}}$ and partition $\\boldsymbol{{\\mathcal{Q}}_\\bullet}$.} \\quad\nLet ${\\mathcal{Q}}_\\bullet=\\{Q_{\\bullet,1},\\dots,Q_{\\bullet,n_\\bullet}\\}$ be the partition of $\\Gamma$ into compact and connected segments $Q_{\\bullet,j}=\\gamma(\\widehat{Q}_{\\bullet,j})$ with $\\widehat{Q}_{\\bullet,j}=[\\widehat{z}_{\\bullet,j-1},\\widehat{z}_{\\bullet,j}]$.\n\n{\\bf Local mesh-sizes $\\boldsymbol{h_{Q}}$, $\\boldsymbol{\\widehat{h}_{Q}}$ and $\\boldsymbol{h_\\bullet}$, $\\boldsymbol{\\widehat{h}_\\bullet}$.}\\quad\nFor each element $Q\\in{\\mathcal{Q}}_\\bullet$, let $h_Q:=|Q|$ be its arclength on the physical boundary and $\\widehat{h}_Q=|\\gamma^{-1}(Q|)$ its length in the parameter domain.\nNote that the lengths $h_{Q}$ and $\\widehat{h}_{Q}$ of an element $Q$ are equivalent, and the equivalence constants depend only on $\\gamma$.\nWe define the local mesh-width function $h_\\bullet\\in L^\\infty(\\Gamma)$ by $h_\\bullet|_Q:=h_{Q}$.\nAdditionally, we define $\\widehat{h}_\\bullet\\in L^\\infty(\\Gamma)$ by $\\widehat{h}_\\bullet|_Q:=\\widehat{h}_{Q}$. \n\n{\\bf Local mesh-ratio $ \\boldsymbol{\\widehat{\\kappa}_\\bullet}$.}\\quad\nWe define the {local mesh-ratio} by\n\\begin{align}\\label{eq:meshratio}\n\\widehat{\\kappa}_\\bullet&:=\\max\\set{\\widehat{h}_{Q}\/\\widehat{h}_{Q'}}{{Q},{Q}'\\in{\\mathcal{Q}}_\\bullet \\text{ with } Q\\cap Q'\\neq \\emptyset}.\n\\end{align}\n\n{\\bf Patches $\\boldsymbol{\\pi_\\bullet^m(\\Gamma_0)}$ and $\\boldsymbol{\\Pi_\\bullet^m(\\Gamma_0)}$.}\nFor $\\Gamma_0\\subseteq\\Gamma$, we inductively define patches by\n\\begin{align}\n \\pi_\\bullet^0(\\Gamma_0) := \\Gamma_0,\n \\quad \n \\pi_\\bullet^m(\\Gamma_0) := \\bigcup\\set{Q\\in{\\mathcal{Q}}_\\bullet}{ {Q}\\cap \\pi_\\bullet^{m-1}(\\Gamma_0)\\neq \\emptyset}.\n\\end{align}\nThe corresponding set of elements is defined as\n\\begin{align}\n \\Pi_\\bullet^m(\\Gamma_0) := \\set{Q\\in {\\mathcal{Q}}_\\bullet}{ {Q} \\subseteq \\pi_\\bullet^m(\\Gamma_0)},\n \\quad\\text{i.e.,}\\quad\n \\pi_\\bullet^m(\\Gamma_0) = \\bigcup\\Pi_\\bullet^m(\\Gamma_0).\n\\end{align}\nTo abbreviate notation, we set $\\pi_\\bullet(\\Gamma_0) := \\pi_\\bullet^1(\\Gamma_0)$ and $\\Pi_\\bullet(\\Gamma_0) := \\Pi_\\bullet^1(\\Gamma_0)$.\nIf $\\Gamma_0=\\{z\\}$ for some $z\\in\\Gamma$, we write $\\pi^m_\\bullet(z):=\\pi^m_\\bullet(\\{z\\})$ and $\\Pi_\\bullet^m(z) := \\Pi_\\bullet^m(\\{z\\})$, where we skip the index for $m=1$ as before.\n\n\n\\subsection{Mesh-refinement}\n\\label{section:mesh-refinement}\nWe suppose that we are given fixed initial knots ${\\mathcal K}_0$.\nFor refinement, we use the following strategy.\n\n \\begin{algorithm}\\label{alg:refinement}\n\\textbf{Input:} Knot vector ${\\mathcal K}_\\bullet$, marked nodes $\\mathcal M_\\bullet\\subseteq{\\mathcal N}_\\bullet$, local mesh-ratio $\\widehat\\kappa_0\\ge1$.\n\\begin{enumerate}[\\rm(i)]\n\\item Define the set of marked elements $\\mathcal M_{\\bullet}':=\\emptyset$.\n\\item If both nodes of an element $Q\\in {\\mathcal{Q}}_\\bullet$ belong to $\\mathcal M_\\bullet$, mark $Q$ by adding it to $\\mathcal M_{\\bullet}'$.\n\\item For all other nodes in $\\mathcal M_\\bullet$, increase the multiplicity if it is less or equal to $p-1$. \nOtherwise mark the elements which contain one of these nodes, by adding them to $\\mathcal M_{\\bullet}'$.\n\\item\nRecursively enrich $\\mathcal M_\\bullet'$ by \n$\\mathcal{U}_\\bullet':=\\{Q\\in{\\mathcal{Q}}_\\bullet\\setminus\\mathcal M_\\bullet':$ $\\exists Q'\\in\\mathcal M_\\bullet'\\quad Q'\\cap Q\\neq\\emptyset$ and $\\max\\{\\widehat h_Q\/\\widehat h_{Q'},\\widehat h_{Q'}\/\\widehat h_Q\\}>\\widehat\\kappa_0\\}$\nuntil $\\mathcal{U}_\\bullet'=\\emptyset$. \n\n\\item\nBisect all $ Q\\in \\mathcal M_\\bullet'$ in the parameter domain by inserting the midpoint of $\\gamma^{-1}(Q)$ with multiplicity one to the current knot vector. \n\\end{enumerate}\n\\textbf{Output:} Refined knot vector ${\\mathcal K}_\\circ=:{\\tt refine}({\\mathcal K}_\\bullet,\\mathcal M_\\bullet)$.\n\\end{algorithm}\n\n\nThe optimal 1D bisection algorithm in~step {\\rm(iii)--(iv)} is analyzed in \\cite{meshoneD}.\nClearly, ${\\mathcal K}_\\circ={\\tt refine}({\\mathcal K}_\\bullet,\\mathcal M_\\bullet)$ is finer than ${\\mathcal K}_\\bullet$ in the sense that ${\\mathcal K}_\\bullet$ is a subsequence of ${\\mathcal K}_\\circ$.\nFor any knot vector ${\\mathcal K}_\\bullet$ on $\\Gamma$, we define ${\\tt refine}({\\mathcal K}_\\bullet)$ as the set of all knot vectors ${\\mathcal K}_\\circ$ on $\\Gamma$ such that there exist knot vectors ${\\mathcal K}_{(0)},\\dots,{\\mathcal K}_{(J)}$ and corresponding marked nodes $\\mathcal M_{(0)},\\dots,\\mathcal M_{(J-1)}$ with ${\\mathcal K}_\\circ$ $={\\mathcal K}_{(J)}$ $={\\tt refine}({\\mathcal K}_{(J-1)},\\mathcal M_{(J-1)}),\\dots,{\\mathcal K}_{(1)}$ $={\\tt refine}({\\mathcal K}_{(0)},\\mathcal M_{(0)})$, and ${\\mathcal K}_{(0)}={\\mathcal K}_\\bullet$. \nNote that ${\\tt refine}({\\mathcal K}_\\bullet,\\emptyset)={\\mathcal K}_\\bullet$, wherefore ${\\mathcal K}_\\bullet\\in{\\tt refine}({\\mathcal K}_\\bullet)$. \nWe define the set of all \\textit{admissible} knot vectors on $\\Gamma$ as \n\\begin{align}\n\\mathbb{K}:={\\tt refine}({\\mathcal K}_0).\n\\end{align}\nAccording to \\cite[Theorem~2.3]{meshoneD}, there holds for arbitrary ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ that \n\\begin{align}\\label{eq:two kappa}\n\\widehat h_Q\/\\widehat h_{Q'}\\le 2\\widehat\\kappa_0 \\quad\\text{for all } Q,Q'\\in{\\mathcal{Q}}_\\bullet \\text{ with }Q\\cap Q'\\neq \\emptyset.\n\\end{align}\nIndeed, one can easily show that \n $\\mathbb{K}$ coincides with the set of all knot vectors ${\\mathcal K}_\\bullet$ which are obtained via iterative bisections in the parameter domain and arbitrary knot multiplicity increases, which satisfy \\eqref{eq:two kappa}. \n\n\n\\subsection{B-splines and NURBS}\n\\label{subsec:splines}\nThroughout this subsection, we consider an arbitrary but fixed sequence {$\\widehat{\\mathcal{K}}_\\bullet:=(t_{\\bullet,i})_{i\\in\\mathbb {Z}}$} on ${\\mathbb R}$ with multiplicities $\\#_\\bullet t_{\\bullet,i}$ which satisfy\n$t_{\\bullet,i-1}\\leq t_{\\bullet,i}$ for $i\\in \\mathbb {Z}$ and $\\lim_{i\\to \\pm\\infty}t_{\\bullet,i}=\\pm \\infty$.\nLet $\\widehat{\\mathcal{N}}_\\bullet:=\\set{t_{\\bullet,i}}{i\\in\\mathbb {Z}}=\\set{\\widehat{{z}}_{\\bullet,j}}{j\\in \\mathbb {Z}}$ denote the corresponding set of nodes with $\\widehat{{z}}_{\\bullet,j-1}<\\widehat{{z}}_{\\bullet,j}$ for $j\\in\\mathbb {Z}$.\nThroughout, we use the convention that $(\\cdot)\/0:=0$.\nFor $i\\in\\mathbb {Z}$, the $i$-th \\textit{B-spline} of degree $p$ is defined for $t\\in{\\mathbb R}$ inductively by\n\\begin{align}\n\\begin{split}\n\\widehat B_{\\bullet,i,0}(t)&:=\\chi_{[t_{\\bullet,i-1},t_{\\bullet,i})}(t),\\\\\n\\widehat B_{\\bullet,i,p}(t)&:=\\frac{t-t_{\\bullet,i-1}}{t_{\\bullet,i-1+p}-t_{\\bullet,i-1}} \\widehat B_{\\bullet,i,p-1}(t)+\\frac{t_{\\bullet,i+p}-t}{t_{\\bullet,i+p}-t_{\\bullet,i}} \\widehat B_{\\bullet,i+1,p-1}(t) \\quad \\text{for } p\\in {\\mathbb N}.\n\\end{split}\n\\end{align}\nThe following lemma collects some\nbasic properties of B-splines; see, e.g., \\cite{Boor-SplineBasics}.\n\n\\begin{lemma}\\label{lem:properties for B-splines}\nFor $-\\infty 0$ { resp.\\ continuous for $p=\\#_\\bullet t_{\\bullet,i}$.}\n\\item \\label{item:B-splines local} For $i\\in\\mathbb {Z}$, the B-spline $\\widehat B_{\\bullet,i,p}$ vanishes outside the interval $[t_{\\bullet,i-1},t_{\\bullet,i+p})$. \nIt is positive on the open interval $(t_{\\bullet,i-1},t_{\\bullet,i+p})$.\n\\item \\label{item:B-splines determined} For $i\\in \\mathbb {Z}$, the B-splines $\\widehat B_{\\bullet,i,p}$ is completely determined by the $p+2$ knots $t_{\\bullet,i-1},\\dots,t_{\\bullet,i+p}$, wherefore we also write\n\\begin{align}\n\\widehat B(\\cdot|t_{\\bullet,i-1},\\dots,t_{\\bullet,i+p}):=\\widehat B_{\\bullet,i,p}.\n\\end{align}\n\n\\item\\label{item:B-splines partition} The B-splines of degree $p$ form a (locally finite) partition of unity, i.e.,\n\\begin{equation}\n\\sum_{i \\in\\mathbb {Z}} \\widehat B_{\\bullet,i,p}=1\\quad \\text{on }{\\mathbb R}.\n\\end{equation}\n\\item\\label{item:interpolatoric} For $i\\in \\mathbb {Z}$ with $t_{\\bullet,i-1}0$.\nFor $i\\in \\mathbb {Z}$ and $p\\in {\\mathbb N}_0$, we define the $i$-th \nNURBS by\n\\begin{equation}\n\\widehat R_{\\bullet,i,p}:=\\frac{w_{\\bullet,i} \\widehat B_{\\bullet,i,p}}{\\sum_{k\\in\\mathbb {Z}} w_{\\bullet,k}\\widehat B_{\\bullet,k,p}}.\n\\end{equation}\nNote that the denominator is locally finite and positive.\nFor any $p\\in{\\mathbb N}_0$, we define the spline space as well as the rational spline space \n\\begin{equation}\\label{eq:NURBS space defined} \n\\widehat{\\mathcal{S}}^p(\\widehat{\\mathcal{K}}_\\bullet):={\\rm span} \\,\n\\set{\\widehat B_{\\bullet,i,p}}{i\\in\\mathbb {Z}}\\quad\\text{and}\\quad\n\\widehat{\\mathcal{S}}^p(\\widehat{\\mathcal{K}}_\\bullet,\\mathcal{W}_\\bullet):={\\rm span} \\,\n\\set{\\widehat R_{\\bullet,i,p}}{i\\in\\mathbb {Z}}.\n\\end{equation}\n\n\n\n\\subsection{Ansatz spaces}\n\\label{section:igabem} \n We abbreviate $N_0:=\\#_0 {\\mathcal N}_0$ and suppose that additionally to the initial knots ${\\mathcal K}_0$, $\\mathcal{W}_0=(w_{0,i})_{i=1-p}^{N_0-p}$ are given initial weights with $w_{0,1-p}=w_{0,N_0-p}$.\nTo apply the results of Section~\\ref{subsec:splines}, extend the knot sequence in the parameter domain, i.e., $\\widehat{{\\mathcal K}}_0=(t_{0,i})_{i=1}^{N_0}$ arbitrarily to $(t_{0,i})_{i\\in \\mathbb {Z}}$ with $t_{0,-p}=\\dots=t_{0,0}=a$, $t_{0,i}\\le t_{0,i+1}$, $\\lim_{i\\to \\pm\\infty}t_{0,i}=\\pm \\infty$.\nFor the extended sequence, we also write $\\widehat{\\mathcal{K}}_0$.\nWe define the weight function \n\\begin{align}\n\\widehat W_0:=\\sum_{k=1-p}^{N_0-p} w_{0,k}\\widehat B_{0,k,p}|_{[a,b)}.\n\\end{align}\n\nLet ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ be a knot vector and abbreviate $N_\\bullet:=\\#_\\bullet{\\mathcal N}_\\bullet$.\nOutside of $(a,b]$, we extend the corresponding knot sequence $\\widehat{\\mathcal K}_\\bullet$ as before to guarantee that $\\widehat{\\mathcal K}_0$ forms a subsequence of $\\widehat{\\mathcal K}_\\bullet$.\nVia {knot insertion} from $\\widehat{\\mathcal K}_0$ to $\\widehat{\\mathcal K}_\\bullet$, Lemma~\\ref{lem:properties for B-splines} \\eqref{item:spline basis} proves the existence and uniqueness of weights $\\mathcal{W}_\\bullet=(w_{\\bullet,i})_{i=1-p}^{N_\\bullet-p}$ with\n\\begin{align}\\label{eq:w}\n\\widehat W_0=\\sum_{k=1-p}^{N_0-p} w_{0,k}\\widehat B_{0,k,p}|_{[a,b)}=\\sum_{k=1-p}^{N_\\bullet-p} w_{\\bullet,k} \\widehat B_{\\bullet,k,p}|_{[a,b)}.\n\\end{align}\nBy choosing these weights, we ensure that the denominator of the considered rational splines does not change.\nThese weights are just convex combinations of the initial weights $\\mathcal{W}_0$; see, e.g., \\cite[Section 11]{Boor-SplineBasics}. \nFor $\\widehat w\\in\\mathcal{W}_\\bullet$, this shows that\n\\begin{align}\\label{eq:weights}\nw_{\\min}:=\\min(\\mathcal{W}_0)\\le\\min(\\mathcal{W}_\\bullet)\\le \\widehat w\\le\\max(\\mathcal{W}_\\bullet)\\le \\max(\\mathcal{W}_0)=:w_{\\max}.\n\\end{align}\nMoreover, $\\widehat B_{\\bullet,1-p}(a)=\\widehat B_{\\bullet,N_\\bullet-p}(b-)=1$ from Lemma \\ref{lem:properties for B-splines} \\eqref{item:interpolatoric} implies that $w_{\\bullet,1-p}=w_{\\bullet,N_\\bullet-p}$.\nFinally, we extend $\\mathcal{W}_\\bullet$ arbitrarily to $(w_{\\bullet,i})_{i\\in\\mathbb {Z}}$ with $w_{\\bullet,i}>0$, identify the extension with $\\mathcal{W}_\\bullet$, and set\n\\begin{equation}\n{\\mathcal{S}}^p({\\mathcal{K}}_{\\bullet},\\mathcal{W}_{\\bullet}):=\\set{\\widehat S_\\bullet\\circ\\gamma^{-1}}{\\widehat S_\\bullet \\in \\widehat{\\mathcal{S}}^p(\\widehat{\\mathcal{K}}_\\bullet,\\mathcal{W}_\\bullet}.\n\\end{equation}\nLemma \\ref{lem:properties for B-splines} \\eqref{item:spline basis} shows that this definition does not depend on how the sequences are extended.\nWe define the transformed basis functions \n\\begin{align}\\label{eq:basis def}\n{R}_{\\bullet,i,p}:=\n\\widehat R_{\\bullet,i,p}\\circ\\gamma^{-1}.\n\\end{align}\n\nWe introduce the ansatz space\n\\begin{align}\\label{eq:hypsing X0}\n{\\mathcal X}_\\bullet:=\\set{V_\\bullet\\in{\\mathcal{S}}^p({\\mathcal{K}}_\\bullet,\\mathcal{W}_\\bullet)}{ V_\\bullet(\\gamma(a))=V_\\bullet(\\gamma(b-))}\\subset H^{1}(\\Gamma)\n\\end{align}w\nLemma \\ref{lem:properties for B-splines} \\eqref{item:spline basis} and \\eqref{item:interpolatoric} show that bases of these spaces are given by\n\\begin{align}\\label{eq:hypsing basis}\n\\set{ R_{\\bullet,i,p}}{i=2-p,\\dots,N_\\bullet-p-1}\\cup \\{ R_{\\bullet,1-p,p}+ R_{\\bullet,N_\\bullet-p,p}\\}.\n\\end{align}\nBy Lemma \\ref{lem:properties for B-splines} \\eqref{item:spline basis}, the ansatz spaces are nested, i.e., \n\\begin{align}\\label{eq:nested}\n{\\mathcal X}_\\bullet\\subseteq {\\mathcal X}_\\circ\\quad\\text{for all }{\\mathcal K}_\\bullet\\in\\mathbb{K}\\text{ and } {\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet).\n\\end{align}\n\nWe define $\\phi_\\bullet:=P_\\bullet\\phi$, where $P_\\bullet$ is throughout either the identity or the $L^2$-orthogonal projection onto the space of transformed piecewise polynomials \n\\begin{align}\n\\mathbf{P}^p({\\mathcal{Q}}_\\bullet):=\\set{\\widehat \\Psi_\\bullet\\circ\\gamma^{-1}}{\\widehat\\Psi_\\bullet \\text{ is }\\widehat{\\mathcal{Q}}_\\bullet\\text{-piecewise polynomial of degree } p}.\n\\end{align}\nThe corresponding Galerkin approximation $U_\\bullet\\in{\\mathcal X}_\\bullet$ reads\n\\begin{align}\\label{eq:hypsing Galerkin}\n \\edual{U_\\bullet}{V_\\bullet} = \\dual{g_\\bullet}{V_\\bullet}_\\Gamma \n\\,\\text{ for all }V_\\bullet\\in {\\mathcal X}_\\bullet,\\quad\\text{where }g_\\bullet:=(1\/2-\\mathfrak{K}')\\phi_\\bullet.\n\\end{align}\nWe note that the choice $\\phi_\\bullet:=\\phi$ is only of theoretical interest as it led to instabilities in our numerical experiments, in contrast to the weakly-singular case \\cite{igafaermann,resigabem,gantner17}. \n\n\n\n\n\n\\section{Main result}\\label{sec:main result}\nIn this section, we introduce a novel adaptive algorithm and state its convergence behavior.\n\n\\subsection{Error estimators} \\label{sec:estimators}\nLet ${\\mathcal K}_\\bullet\\in\\mathbb{K}$. \nThe definition of the error estimator \\eqref{eq:hyper estimator} requires the additional regularity \n$\\phi\\in L^2(\\Gamma)$, which leads to \n$g_\\bullet=(1\/2-{\\mathcal K}')\\phi_\\bullet\\in L^2(\\Gamma)$ due to the mapping properties of \n$\\mathfrak{K}'$.\nMoreover, note that \n$\\mathfrak{W} U_\\bullet\\in L^2(\\Gamma)$ due to the mapping properties of \n$\\mathfrak{W}$ and the fact that \n$U_\\bullet\\in{\\mathcal X}_\\bullet\\subset H^1(\\Gamma)$.\nTherefore, the following error indicators are well-defined.\nWe consider the sum of weighted-residual error indicators by \\cite{cs95,carstensen97} and oscillation terms\n\\begin{subequations}\\label{eq:hyper estimator}\\begin{align}\n\\eta_\\bullet(z)^2:={\\rm res}_\\bullet(z)^2+{\\rm osc}_\\bullet(z)^2 \\quad\\text{for all }z\\in{\\mathcal N}_\\bullet,\n\\end{align}\nwhere\n\\begin{align}\n{\\rm res}_\\bullet(z):=\\norm{h_\\bullet^{1\/2}(g_\\bullet-\\mathfrak{W} U_\\bullet)}{L^2(\\pi_\\bullet(z))} \\quad \\text{and}\\quad {\\rm osc}_\\bullet(z):=\\norm{h_\\bullet^{1\/2}(\\phi-\\phi_\\bullet)}{L^2(\\pi_\\bullet(z))}. \n\\end{align}\n\\end{subequations}\nRecall the convention~\\eqref{eq:estimator convention} for $\\alpha_\\bullet\\in \\{\\eta_\\bullet,{\\rm res}_\\bullet,{\\rm osc}_\\bullet\\}$.\n\nTo incorporate the possibility of knot multiplicity decrease, we define the knots ${\\mathcal K}_{\\bullet\\ominus 1}$ by decreasing the multiplicities of all nodes $z\\in{\\mathcal N}_\\bullet$ whose multiplicity is larger than $1$ and the original multiplicity $\\#_0 z$ if $z\\in{\\mathcal N}_0$, i.e., ${\\mathcal N}_{\\bullet\\ominus 1}:={\\mathcal N}_\\bullet$ and $\\#_{\\bullet\\ominus 1}z:=\\max\\{\\#_\\bullet z-1,1,\\#_0z\\}$ (where we set $\\#_0z:=0$ if $z\\notin{\\mathcal N}_0$).\nLet \n\\begin{align}\nJ_{\\bullet\\ominus1}:L^2(\\Gamma)\\to{\\mathcal X}_{\\bullet\\ominus 1}\n\\end{align}\ndenote the corresponding Scott--Zhang-type projection \nfrom Section~\\ref{section:scott} below.\nTo measure the approximation error by multiplicity decrease, we consider the following indicators \n\\begin{align}\\label{eq:inverse estimator}\n\\mu_\\bullet(z):=\\norm{ h_\\bullet^{-1\/2}(1-J_{\\bullet\\ominus 1})U_\\bullet}{L^2(\\pi_\\bullet^{2p+1}(z))}\\text{ for all }z\\in{\\mathcal N}_\\bullet.\n\\end{align}\nWe define $\\mu_\\bullet$ and $\\mu_\\bullet({\\mathcal{S}}_\\bullet)$ as in \\eqref{eq:hyper estimator}.\n\n\\subsection{Adaptive algorithm}\nWe propose the following adaptive algorithm.\n\\begin{algorithm}\\label{the algorithm}\n\\textbf{Input:} Adaptivity parameters $0<\\theta<1$, $\\vartheta\\ge0$, $C_{\\rm min}\\ge 1$, $C_{\\rm mark}>0$.\\\\\n\\textbf{Adaptive loop:} For each $\\ell=0,1,2,\\dots$ iterate the following steps {\\rm(i)--(iv)}:\n\\begin{itemize}\n\\item[\\rm(i)] Compute Galerkin approximation\n$U_\\ell\\in{\\mathcal X}_\\ell$.\n\\item[\\rm(ii)] Compute refinement and coarsening indicators $\\eta_{\\ell}(z)$ and $\\mu_\\ell(z)$ for all $z\\in{\\mathcal N}_{\\ell}$.\n\\item[\\rm(iii)] Determine an up to the multiplicative constant \n$C_{\\rm min}$ \nminimal set of nodes $\\mathcal M_\\ell^1\\subseteq{\\mathcal N}_\\ell$, which satisfies the D\\\"orfler marking \n\\begin{align}\\label{eq:Doerfler}\n \\theta\\,\\eta_{\\ell}^2 \\le \\eta_\\ell(\\mathcal M_\\ell^1)^2.\n\\end{align}\n\\item[\\rm(iv)]Determine a set of nodes $\\mathcal M^-_{\\ell}\\subseteq \\set{z\\in{\\mathcal N}_\\ell}{\\#_\\ell z>\\#_{\\ell\\ominus 1}z}$ with $|\\mathcal M^-_\\ell| \\le C_{\\rm mark}|\\mathcal M_\\ell|$, which satisfies the following marking criterion \n\\begin{align}\\label{eq:invdo}\n\\mu_{\\ell}(\\mathcal M_\\ell^-)^2\\le \\vartheta \\eta_\\ell^2, \n\\end{align}\nand define $\\mathcal M_\\ell^2:=\\set{z\\in{\\mathcal N}_\\ell}{\\pi_\\ell(z)\\cap\\mathcal M_\\ell^{-}\\neq\\emptyset}$ as well as $\\mathcal M_\\ell:=\\mathcal M_\\ell^1\\cup\\mathcal M_\\ell^2$.\n\\item[\\rm(v)] Generate refined intermediate knot vector ${\\mathcal K}_{\\ell+1\/2}:={\\tt refine}({\\mathcal K}_\\ell,\\mathcal M_\\ell)$ and then coarsen knot vector ${\\mathcal K}_{\\ell+1\/2}$ to ${\\mathcal K}_{\\ell+1}$ from by decreasing the multiplicities of all $z\\in\\mathcal M^-_\\ell$ by one.\n\\end{itemize}\n\\textbf{Output:} Approximations \n$U_\\ell$ and error estimators $\\eta_\\ell$ for all $\\ell \\in {\\mathbb N}_0$.\n\\end{algorithm}\n\n\\begin{remark}\\label{rem:feature}\n{\\rm (a)} By additionally marking the nodes $\\mathcal M_\\ell^2$, we enforce that the neighboring elements of any node $z\\in\\mathcal M_\\ell^-$, marked for multiplicity decrease, are bisected.\nWe emphasize that the enriched set $\\mathcal M_\\ell$ still satisfies the D\\\"orfler marking with parameter $\\theta$ and is minimal up to the multiplicative constant $C_{\\rm min}+3C_{\\rm mark}$.\n\n{\\rm (b)} Algorithm~\\ref{the algorithm} allows the choice $\\vartheta=0$ and $\\mathcal M_\\ell^-=\\emptyset$, and then formally coincides with the adaptive algorithm from \\cite{resigabem} for the weakly-singular integral equation.\n\n{\\rm (c)} \nLet even $\\const{min}\\ge 3$.\nIf we choose in each step $\\widetilde\\mathcal M_\\ell^1$ up to the multiplicative constant $\\const{min}\/3$ minimal such that $\\theta\\,\\eta_{\\ell}^2 \\le \\eta_\\ell(\\widetilde\\mathcal M_\\ell^1)^2$, and define $\\mathcal M_\\ell^1:=\\set{z\\in{\\mathcal N}_\\ell}{\\pi_\\ell(z)\\cap\\widetilde\\mathcal M_\\ell^1\\neq\\emptyset}$, then $\\mathcal M_\\ell^1$ is as in Algorithm~\\ref{the algorithm} {\\rm (iii)}, then this leads to standard $h$-refinement with no multiplicity increase and thus no decrease (independently on how $\\const{mark}$ and $\\vartheta$ are chosen).\n\n\\end{remark}\n\n\n\\subsection{Linear and optimal convergence}\n\\label{sec:main}\nOur main result is that Algorithm~\\ref{the algorithm} guarantees linear convergence with optimal algebraic rates.\nFor standard BEM with piecewise polynomials, such a result is proved in \\cite{gantumur,fkmp,part1} for weakly-singular integral equations and in \\cite{gantumur,part2} for hyper-singular integral equations, where \\cite{part1,part2} also account for data oscillation terms.\nFor IGABEM for the weakly-singular integral equation (but without knot multiplicity decrease), an analogous result is already proved in our recent work \\cite{optigabem}.\nTo precisely state the main theorem, let\n\\begin{align}\n\\mathbb{K}(N):=\\set{{\\mathcal K}_\\bullet\\in\\mathbb{K}}{\\#_\\bullet{\\mathcal N}_\\bullet-\\#_0{\\mathcal N}_0\\le N}\n\\end{align}\nbe the finite set of all refinements having at most $N$ knots more than ${\\mathcal K}_0$.\n\nAnalogously to \\cite{axioms}, we introduce the estimator-based approximability constant\n\\begin{align}\\label{eq:Capprox}\n\\norm{u}{\\mathbb{A}_s}:=\\sup_{N\\in{\\mathbb N}_0} \\big( (N+1)^s \\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}(N)} \\eta_\\bullet\\big)\\in\\mathbb{R}_{\\ge 0} \\cup\\{\\infty\\}\\quad \\text{for all }s>0. \n\\end{align}\nBy this constant, one can characterize the best possible convergence rate.\nIn explicit terms, this constant is finite if and only if an algebraic convergence rate of $\\mathcal{O}(N^{-s})$ for the estimator is possible for suitably chosen knot vectors.\nSimilarly, we define \n\\begin{align}\\label{eq:Ks}\n\\begin{split}\n&\\mathbb{K}^1:=\\set{{\\mathcal K}_\\bullet\\in\\mathbb{K}}{\\#_\\bullet z_{\\bullet,j}=\\max\\{1,\\#_0 z_{\\bullet,j}\\}\\text{ for all }j=1,\\dots,|{\\mathcal N}_\\bullet|-1},\\\\\n&\\mathbb{K}^1(N):=\\mathbb{K}(N)\\cap\\mathbb{K}^1,\n\\\\\n&\\mathbb{K}^p:=\\set{{\\mathcal K}_\\bullet\\in\\mathbb{K}}{\\#_\\bullet z_{\\bullet,j}=p\\text{ for all }j=1,\\dots,|{\\mathcal N}_\\bullet|-1},\\\\\n&\\mathbb{K}^p(N):=\\set{{\\mathcal K}_\\bullet\\in\\mathbb{K}^p}{\\#_\\bullet{\\mathcal N}_\\bullet-\\#_{0,p}{\\mathcal N}_0\\le N} \\text{ with }{\\mathcal K}_{0,p}\\in\\mathbb{K}^p\\text{ and }{\\mathcal N}_{0,p}={\\mathcal N}_0,\n\\end{split}\n\\end{align}\nand\n\\begin{align}\n\\norm{u}{\\mathbb{A}^1_s}:=\\sup_{N\\in{\\mathbb N}_0} \\big( (N+1)^s \\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}^1(N)} \\eta_\\bullet\\big)\\quad\\text{ and }\\quad\n\\norm{u}{\\mathbb{A}^p_s}:=\\sup_{N\\in{\\mathbb N}_0} \\big( (N+1)^s \\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}^p(N)} \\eta_\\bullet\\big).\n\\end{align}\nThe constant $\\norm{u}{\\mathbb{A}^1_s}$ characterizes the best possible convergence rate starting from ${\\mathcal K}_0$ when only bisection is used and all new nodes have multiplicity $1$.\nThe constant $\\norm{u}{\\mathbb{A}^p_s}$ characterizes the best possible rate starting from the coarsest knot vector ${\\mathcal K}_{0,p}\\in\\mathbb{K}^p$ when only bisection is used and all new nodes have maximal multiplicity $p$. \nHence, $\\norm{u}{\\mathbb{A}^p_s}$ characterizes the rate for standard BEM with continuous piecewise polynomial ansatz functions.\nNote that the constants coincide if $p=1$.\n\nThe following theorem is the main result of our work.\nThe proof is given in Section~\\ref{sec:proof}.\n\n\\begin{theorem}\\label{thm:main}\nLet \n$\\phi\\in L^2(\\Gamma)$ \nso that the weighted-residual error estimator is well-defined.\nThen, the estimator $\\eta$ from \\eqref{eq:hyper estimator} is reliable as well as weakly efficient, i.e., there exist $C_{\\rm rel},C_{\\rm eff}>0$ such that, for all ${\\mathcal K}_\\bullet\\in\\mathbb{K}$, \n\\begin{align}\\label{eq:reliable}\nC_{\\rm rel}^{-1}\\norm{u-U_\\bullet}{\\widetilde{H}^{1\/2}(\\Gamma)}\n\\le \n\\eta_\\bullet\n\\le\nC_{\\rm eff} \\big( \\norm{h_\\bullet^{1\/2} \\partial_\\Gamma( u - U_\\bullet )}{L^2(\\Gamma)}^2\n + \\norm{h_\\bullet^{1\/2} ( \\phi - \\phi_\\bullet )}{L^2(\\Gamma)}^2 \\big)^{1\/2}.\n\\end{align}\nFor each $0<\\theta\\le1$, there is a constant $\\vartheta_{\\rm opt}>0$ such that for all $0\\le \\vartheta<\\vartheta_{\\rm opt}$ there exist constants $0<\\q{conv}<1$ and $C_{\\rm conv}>0$ such that Algorithm~\\ref{the algorithm} is linearly convergent in the sense that\n\\begin{align}\\label{eq:R-linear}\n\\eta_{\\ell+k}\\leq C_{\\rm conv} \\,\\q{conv}^k\\,\\eta_\\ell \\quad \\text{for all }k,\\ell\\in{\\mathbb N}_0.\n\\end{align}\nMoreover, there is a constant $0<\\theta_{\\rm opt}<1$ such that for all $0<\\theta<\\theta_{\\rm opt}$ and $0\\le\\vartheta<\\vartheta_{\\rm opt}$, there exist constants $c_{\\rm opt},C_{\\rm opt}>0$ such that, for all $s>0$, there holds that \n\\begin{align}\\label{eq:optimal}\nc_{\\rm opt}\\norm{u}{\\mathbb{A}_s}\n\\le\n\\sup_{\\ell\\in{\\mathbb N}_0}{(\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0+1)^{s}}{\\eta_\\ell}\\leC_{\\rm opt} \\norm{u}{\\mathbb{A}_s}.\n\\end{align}\nFinally, there exist constants $c_{\\rm apx},\\const{apx}>0$ such that, for all $s>0$, there holds that \n\\begin{align}\\label{eq:classes}\nc_{\\rm apx}\\norm{u}{\\mathbb{A}_s^1}\\le\\norm{u}{\\mathbb{A}_s}\\le \n\\min\\{\\norm{u}{\\mathbb{A}_s^1},\\norm{u}{\\mathbb{A}_s^p}\\}\\le\\const{apx}\\norm{u}{\\mathbb{A}_s^1}. \n\\end{align}\n\nThe constants $C_{\\rm rel}$ and $C_{\\rm eff}$ depend only on $\\gamma, p, \\widehat{\\mathcal{Q}}_0, w_{\\min},$ and $w_{\\max}$.\nThe constant $\\vartheta_{\\rm opt}$ depends additionally on $\\theta$.\nThe constants $\\q{conv}$ as well as $C_{\\rm conv}$ depend further on $\\theta$ and $\\vartheta$.\nThe constant $\\theta_{\\rm opt}$ depends only on $\\gamma, p, \\widehat{\\mathcal{Q}}_0, w_{\\min},$ and $w_{\\max}$, whereas, $C_{\\rm opt}$ depends additionally on $\\theta,\\vartheta,C_{\\rm min}, C_{\\rm mark},$ and $s$.\nThe constant $c_{\\rm opt}$ depends only on $\\#_0{\\mathcal N}_0$.\nFinally, the constants $c_{\\rm apx},\\const{apx}$ depend only on $\\gamma, p, \\widehat{\\mathcal{Q}}_0, w_{\\min}$, $w_{\\max}$, and $s$.\n\\hfill$\\square$\n\\end{theorem}\n\n\\begin{remark}\nTheorem~\\ref{thm:main} holds accordingly for indirect BEM, where\n $g=(1\/2-\\mathfrak{K}')\\phi$ in \\eqref{eq:hyper strong} is replaced by $g=\\phi$, and $g_\\bullet=(1\/2-\\mathfrak{K}')\\phi_\\bullet$ in~\\eqref{eq:hypsing Galerkin} is replaced either by $g_\\bullet=\\phi$ or by $g_\\bullet=\\phi_\\bullet$. \nIndeed, due to the absence of the operator $\\mathfrak{K}'$ for indirect BEM, the proof is even simplified.\n\\end{remark}\n\n\\section{Proof of Theorem~\\ref{thm:main} }\n\\label{sec:proof}\n\nTo prove Theorem~\\ref{thm:main}, we follow the abstract convergence theory for adaptive algorithms of \\cite{axioms}, which provide a set of so-called \\textit{axioms of adaptivity}, \nwhich automatically guarantee linear convergence at optimal algebraic rate. \nAlthough we cannot directly apply their result, since it does not cover multiplicity increase or decrease, we will verify slightly modified axioms, which yield Theorem~\\ref{thm:main} with the same ideas as in \\cite{axioms}.\nIn Section~\\ref{subsec:abstract}, we present these axioms.\nTheir verification, which is inspired by the corresponding verification for standard BEM \\cite{part2}, is postponed to Section~\\ref{sec:hypsing stability}--\\ref{sec:hypsing discrete reliability} and \\ref{subsec:hypsing orthogonality}--\\ref{sec:hypsing closure}, after providing some auxiliary results in Section~\\ref{sec:interpolation}--\\ref{sec:inverse inequalities}.\nIn Section~\\ref{sec:reliable hyper} and \\ref{sec:linear convergence}--\\ref{sec:optimal convergence}, we briefly show how these axioms conclude \nreliability in \\eqref{eq:reliable}, linear convergence \\eqref{eq:R-linear}, and optimal convergence \\eqref{eq:optimal}\n(along the lines of \\cite{axioms}).\nEfficiency in \\eqref{eq:reliable} is proved in Section~\\ref{sec:efficient hyper} similarly as for standard BEM \\cite[Section~3.2]{invest}. \nFinally, Section~\\ref{sec:classes} verifies the relation \\eqref{eq:classes} between the approximability constants. \n\n\n\\subsection{Axioms of adaptivity}\\label{subsec:abstract}\nIn this section, we formulate node-based versions of the \\textit{axioms of adaptivity} of \\cite{axioms}. \nThese are not satisfied for the error estimator $\\eta$ itself, but only for a locally equivalent estimator $\\widetilde\\eta$.\nTo introduce this estimator, we first recall an equivalent mesh-size function that has been constructed in \\cite[Proposition~5.8.2]{gantner17} or in \\cite[Proposition~4.2]{optigabem} in a slightly different element-based version.\n\n\n\\begin{proposition}\\label{prop:h tilde}\nFor ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ and $z\\in{\\mathcal N}_\\bullet$, \nlet $z_{\\bullet,{\\rm left}}\\in{\\mathcal N}_\\bullet\\cap\\pi_\\bullet(z)$ be the (with respect to $\\gamma$) left neighbor and $z_{\\bullet,{\\rm right}}\\in {\\mathcal N}_\\bullet\\cap\\pi_\\bullet(z)$ the right neighbor of $z$. \nLet $\\#_\\bullet z_{\\bullet, \\rm left},\\#_\\bullet z_{\\bullet,\\rm right}$ be the corresponding multiplicities. \nThen, there exist $0<\\ro{eq}<1$ and $\\const{eq}>0$ such that\n\\begin{align}\\label{eq:tilde h equivalent}\n\\const{eq}^{-1}h_\\bullet|_{\\pi_\\bullet(z)}\\le\n\\widetilde h_{\\bullet,z}\n:=|\\gamma^{-1}(\\pi_\\bullet(z))| \\, \\ro{eq}^{\\#_\\bullet z_{\\bullet,{\\rm left}}+\\#_\\bullet z+\\#_\\bullet z_{\\bullet,{\\rm right}}}\n\\le \\const{eq} h_\\bullet|_{\\pi_\\bullet(z)},\n\\end{align}\nwhere $\\ro{eq}$ depends only on $p$ and $\\widehat{\\mathcal{Q}}_0$ and $\\const{eq}$ depends additionally on $\\gamma$.\nIf additionally ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$, then there exists a constant $0<\\ro{ctr}<1$ such that for all $ z\\in{\\mathcal N}_\\bullet$, whose patch is changed by bisection or multiplicity increase (i.e., $\\pi_\\bullet(z)\\neq\\pi_\\circ(z)$ or $\\#_\\bullet z_{\\bullet,{\\rm left}}\\neq\\#_\\bullet z_{\\circ,{\\rm left}}$ or $\\#_\\bullet z\\neq\\#_\\circ z$ or $\\#_\\bullet z_{\\bullet,{\\rm right}}\\neq\\#_\\circ z_{\\bullet,{\\rm right}}$), \nand all $z'\\in{\\mathcal N}_\\circ$, \nit holds that\n\\begin{align}\\label{eq:h tilde ctr}\n\\widetilde h_{\\circ,z'}\\le\\ro{ctr}\\,\\widetilde h_{\\bullet,z}\\quad\\text{if }z'\\in \\pi_\\bullet(z)\\setminus{\\mathcal N}_\\bullet\\quad\\text{ or }\\quad\\text{if }z'=z.\n\\end{align}\nwhere $\\ro{ctr}$ depends only on $p$ and $\\widehat{\\mathcal{Q}}_0$. \\hfill$\\square$\n\\end{proposition}\n\nFor ${\\mathcal K}_\\bullet\\in\\mathbb{K}$, we define the estimator\n\\begin{subequations}\\label{eq:hyper estimator tilde}\n\\begin{align}\n\\widetilde\\eta_\\bullet(z)^2:=\\widetilde{\\rm res}_\\bullet(z)^2+\\widetilde{\\rm osc}_\\bullet(z)^2 \\quad\\text{for all }z\\in{\\mathcal N}_\\bullet,\n\\end{align}\nwhere\n\\begin{align}\n\\widetilde{\\rm res}_\\bullet(z):=\\widetilde h_{\\bullet,z}^{\\,1\/2}\\norm{g_\\bullet-\\mathfrak{W} U_\\bullet}{L^2(\\pi_\\bullet(z))}\\quad\\text{and}\\quad \\widetilde{\\rm osc}_\\bullet(z):=\\widetilde h_{\\bullet,z}^{\\,1\/2}\\norm{\\phi-\\phi_\\bullet}{L^2(\\pi_\\bullet(z))}.\n\\end{align}\n\\end{subequations}\nIn particular, \\eqref{eq:tilde h equivalent} implies the local equivalence \n\\begin{align}\\label{eq:abstract equivalence}\n\\const{eq}^{-1}\\eta_\\bullet(z)^2\\le \\widetilde\\eta_\\bullet(z)^2\\le\\const{eq}\\eta_\\bullet(z)^2\\quad\\text{for all }z\\in{\\mathcal N}_\\bullet.\n\\end{align}\n\nTo present the axioms of adaptivity in a compact way, we abbreviate for ${\\mathcal K}_\\bullet,{\\mathcal K}_\\circ\\in\\mathbb{K}$ the corresponding perturbation terms\n\\begin{align}\n\\varrho_{\\bullet,\\circ}:=\\norm{U_\\bullet-U_\\circ}{H^{1\/2}(\\Gamma)}+\\norm{\\phi_\\bullet-\\phi_\\circ}{H^{-1\/2}(\\Gamma)}.\n\\end{align}\nMoreover, we define the set of all nodes in ${\\mathcal N}_\\bullet\\cap {\\mathcal N}_\\circ$ whose patch is identical in ${\\mathcal K}_\\bullet$ and ${\\mathcal K}_\\circ$\n\\begin{align}\\label{eq:nnr}\n\\begin{split}\n{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}:=\\big\\{z\\in{\\mathcal N}_\\bullet\\cap {\\mathcal N}_\\circ:\\,&\\pi_\\bullet(z)=\\pi_\\circ(z), \n\\#_\\bullet z'=\\#_\\circ z' \\text{ for all } z'\\in\\{z,z_{\\bullet,\\rm left},z_{\\bullet,\\rm right}\\}\\big\\}.\n\\end{split}\n\\end{align}\nWe abbreviate its complement in ${\\mathcal N}_\\bullet$ and ${\\mathcal N}_\\circ$\n\\begin{align}\n{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}:={\\mathcal N}_\\bullet\\setminus{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}\\quad\\text{and}\\quad{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}:={\\mathcal N}_\\circ\\setminus{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}. \n\\end{align}\nIn Section~\\ref{sec:hypsing stability}--\\ref{sec:hypsing discrete reliability}, we will verify that if $\\vartheta>0$ is chosen sufficiently small, then there exist constants $C_{\\rm stab},C_{\\rm red},C_{\\rm qo}, C_{\\rm ref},C_{\\rm drel}, \\const{son},C_{\\rm clos}\\ge 1$ and $0\\le\\q{red},\\varepsilon_{\\rm qo}<1$ \n such that the following properties for the estimator (E1)--(E4) and the refinement (R1)--(R3) are satisfied:\n\n\\begin{itemize}\n\\item[(E1)]\\textbf{Stability on non-refined node patches:} For all ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ and all ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$ as well as for all $\\ell\\in{\\mathbb N}_0$, ${\\mathcal K}_\\bullet:={\\mathcal K}_\\ell$, and ${\\mathcal K}_\\circ:={\\mathcal K}_{\\ell+1}$, it holds that\n\\begin{align*}\n|\\widetilde\\eta_\\circ({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})-\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})|\\leC_{\\rm stab}\\,\\varrho_{\\bullet,\\circ}.\n\\end{align*}\n\\item[(E2)]\\textbf{Reduction on refined node patches:} For all ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ and all ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$ as well as for all $\\ell\\in{\\mathbb N}_0$, ${\\mathcal K}_\\bullet:={\\mathcal K}_\\ell$ and ${\\mathcal K}_\\circ:={\\mathcal K}_{\\ell+1}$, it holds that\n\\begin{align*}\n\\widetilde\\eta_\\circ({\\mathcal N}^{\\rm ref}_{\\circ,\\bullet})^2\\le\\q{red}\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ})^2+C_{\\rm red}^2\\,\\varrho_{\\bullet,\\circ}^2.\n\\end{align*}\n\\item[(E3)] \\textbf{Discrete reliability:} For all ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ and all ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$,\nthere exists ${\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}\\subseteq\\RR_{\\bullet,\\circ}\\subseteq{\\mathcal N}_\\bullet$ with\n $\\#_\\bullet\\RR_{\\bullet,\\circ}\\leC_{\\rm ref} (\\#_\\circ{\\mathcal N}_\\circ-\\#_\\bullet{\\mathcal N}_\\bullet)$ such that\n\\begin{align*}\n\\varrho_{\\bullet,\\circ}^2\\le\nC_{\\rm drel}^2\\,\\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ})^2.\n\\end{align*}\n\\item[(E4)] \\textbf{General quasi-orthogonality:} There holds that\n\\begin{align*}\n0\\le\\varepsilon_{\\rm qo}<\\sup_{\\delta>0}\\frac{1-(1+\\delta)(1-(1-\\rho_{\\rm red})\\theta)}{C_{\\rm red}+(2+\\delta^{-1})C_{\\rm stab}^2},\n\\end{align*}\nand the sequence of knots $({\\mathcal K}_\\ell)_{\\ell\\in{\\mathbb N}_0}$ satisfies \nthat \n\\begin{align*}\n\\sum_{k=\\ell}^{\\ell+N}(\\varrho_{k,k+1}^2-\\varepsilon_{\\rm qo}\\widetilde\\eta_k^{\\,2})\\leC_{\\rm qo}\\, \\widetilde\\eta_\\ell^{\\,2}\\quad \\text{for all }\\ell,N\\in{\\mathbb N}_0. \n\\end{align*}\n\\item[(R1)]\\textbf{Son estimate}\nFor all $\\ell\\in{\\mathbb N}_0$, it holds that \n\\begin{align*}\n\\#_{\\ell+1}{\\mathcal N}_{\\ell+1}\\le \\const{son} \\#_\\ell{\\mathcal N}_\\ell.\n\\end{align*}\n\n\\item[(R2)] \\textbf{Closure estimate:} \nFor all $\\ell\\in{\\mathbb N}_0$, there holds that \n\\begin{align*}\n\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0\\leC_{\\rm clos} \\sum_{k=0}^{\\ell-1}\\#_k\\mathcal M_k.\n\\end{align*}\n\\item[(R3)] \\textbf{Overlay property:} \nFor all ${\\mathcal K}_\\bullet, {\\mathcal K}_\\star\\in\\mathbb{K}$, there exists a common refinement ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)\\cap{\\tt refine}({\\mathcal K}_\\star)$ such that \n\\begin{align*}\n\\#_\\circ{\\mathcal N}_\\circ \\le \\#_\\bullet{\\mathcal N}_\\bullet+\\#_{\\star}{\\mathcal N}_{\\star}-\\#_0{\\mathcal N}_0.\n\\end{align*}\n\\end{itemize}\n\n\n\\subsection{Interpolation theory}\n\\label{sec:interpolation}\nWe start with a maybe well-known abstract interpolation result (stated, e.g., in {\\cite[Lemma~2]{hypsing3d}}), which will be applied in the following. \n\\begin{lemma}\\label{lem:discrete interpolation}\nFor $j=0,1$, let $H_j$ be Hilbert spaces with subspaces $X_j\\subseteq H_j$, which satisfy the continuous inclusions $H_0\\supseteq H_1$ and $X_0\\supseteq X_1$.\nAssume that $A:H_j\\to X_j$ is a well-defined linear and continuous projection with operator norm $c_j=\\norm{A:H_j\\to X_j}{}$, for both $j=0,1$. \nThen, there holds equivalence of the interpolation norms\n\\begin{align}\n\\norm{v}{[H_0,H_1]_\\sigma}\\le \\norm{v}{[X_0,X_1]_\\sigma}\\le c_0^{1-\\sigma} c_1^\\sigma \\norm{v}{[H_0,H_1]_\\sigma}\\quad \\text{for all } v\\in [X_0,X_1]_\\sigma\n\\end{align}\nand all $0<\\sigma<1$.\\hfill$\\square$\n\\end{lemma}\nIn the following, we write $(X,\\norm{\\cdot}{X})\\simeq (Y,\\norm{\\cdot}{Y})$ if $X=Y$ in the sense of sets with equivalent norms $\\norm{\\cdot}{X}\\simeq\\norm{\\cdot}{Y}$. The Stein-Weiss interpolation theorem (see, e.g., \\cite[Theorem 5.4.1]{bergh}) shows for fixed $\\alpha\\in\\{0,1\\}$ that for ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ there holds that\n\\begin{align}\\label{eq:intp1}\n(L^2(\\Gamma),\\norm{h_\\bullet^{\\alpha-\\sigma}(\\cdot)}{L^2(\\Gamma)})\\simeq [(L^2(\\Gamma),\\norm{h_\\bullet^{\\alpha}(\\cdot)}{L^2(\\Gamma)}),(L^2(\\Gamma),\\norm{h_\\bullet^{\\alpha-1}(\\cdot)}{L^2(\\Gamma)})]_\\sigma,\n\\end{align}\nwhere the hidden constants depend only on $\\Gamma$ and $\\sigma$.\nMoreover, it holds by definition that\n\\begin{align}\\label{eq:intp2}\n(H^\\sigma(\\Gamma), \\norm{\\cdot}{H^\\sigma(\\Gamma)}) = [(L^2(\\Gamma),\\norm{\\cdot}{L^2(\\Gamma)}),(H^1(\\Gamma), \\norm{\\cdot}{H^1(\\Gamma)}]_\\sigma.\n\\end{align}\n\n\n\n\\subsection{Scott--Zhang-type projection}\n\\label{sec:scott zhang}\n\\label{section:scott} \n\\noindent\nIn this section, we introduce a Scott--Zhang-type operator for ${\\mathcal K}_\\bullet\\in\\mathbb{K}$.\nIn \\cite[Section 2.1.5]{overview}, it is shown that, for $i\\in \\{1-p,\\dots,N_\\bullet-p\\}$, there exist dual basis {functions $\\widehat B_{\\bullet,i,p}^*\\in L^2(a,b)$ such that\n\\begin{align}\n{\\rm supp} \\widehat B_{\\bullet,i,p}^*&={\\rm supp} \\widehat B_{\\bullet,i,p}=[t_{\\bullet,i-1},t_{\\bullet,i+p}],\\\\\n\\int_a^b \\widehat B_{\\bullet,i,p}^* \\widehat B_{\\bullet,j,p} \\,dt&=\\delta_{ij}=\\begin{cases} 1, \\text{ if }i=j,\\\\ 0,\\text{ else,}\\end{cases}\\\\\n\\label{eq:dual inequality}\n\\norm{\\widehat B_{\\bullet,i,p}^*}{L^2([a,b])}&\\le 9^p(2p+3) {|{\\rm supp} \\widehat B_{\\bullet,i,p}|^{-1\/2}}.\n\\end{align}\nEach dual basis function depends only on the knots $t_{\\bullet,i-1},\\dots,t_{\\bullet,i+p}$. \nWith the denominator $\\widehat w$ from \\eqref{eq:w}, define \n\\begin{align}\\label{eq:ripstar}\n\\widehat R_{\\bullet,i,p}^*:=\\widehat B_{\\bullet,i,p}^* \\widehat w\/w_{\\bullet,i}.\n\\end{align}\nThis immediately proves that \n\\begin{align}\n\\label{eq:duality}&\\int_a^b \\widehat R_{\\bullet,i,p}^* (R_{\\bullet,j,p}\\circ\\gamma) dt=\\delta_{ij}=\\begin{cases} 1,\\quad \\text{if }i=j,\\\\ 0,\\quad \\text{else,}\\end{cases}\n\\end{align}\nand\n\\begin{align}\\label{eq:dual inequality2}\n\\norm{\\widehat R_{\\bullet,i,p}^*}{L^2(a,b)}\\lesssim 9^p(2p+3) {|{\\rm supp} R_{\\bullet,i,p}|^{-1\/2}},\n\\end{align}\nwhere the hidden constant depends only on $\\gamma, w_{\\min}$, and $w_{\\max}$. \nWith the abbreviation $N_\\bullet:=\\#_\\bullet {\\mathcal N}_\\bullet$, we define the Scott--Zhang-type operator $J_\\bullet:L^2(\\Gamma)\\to{\\mathcal X}_\\bullet$ by\n\\begin{subequations}\\label{eq:scotty}\n\\begin{align}\n\\begin{split}\nJ_\\bullet v&:=\n\\Big(\\int_a^b \\frac{\\widehat R_{\\bullet,1-p,p}^*+\\widehat R_{\\bullet,N_\\bullet-p,p}^*}{2}( v\\circ\\gamma) \\,dt \\Big)(R_{\\bullet,1-p,p}+R_{\\bullet,N_\\bullet-p,p})\\\\\n&\\quad+\\sum_{i=1-p}^{N_\\bullet-p-1} \\Big(\\int_a^b \\widehat R_{\\bullet,i,p}^* (v \\circ\\gamma) \\,dt \\Big)R_{\\bullet,i,p}.\n\\end{split}\n\\end{align}\n\\end{subequations}\nA similar operator, namely $I_\\bullet := \\sum_{i=1-p}^{N_\\bullet-p} \\Big(\\int_a^b \\widehat R_{\\bullet,i,p}^* (v\\circ \\gamma) \\,dt \\Big)R_{\\bullet,i,p}$, has been analyzed in \\cite[Section 3.1.2]{overview}.\nHowever, $I_\\bullet$ is not applicable here because it does not guarantee that $I_\\bullet v$ is continuous at $\\gamma(a)=\\gamma(b)$.\n\n\\begin{proposition}\\label{lem:Scott properties}\nGiven ${\\mathcal K}_\\bullet\\in \\mathbb{K}$, the operator $J_\\bullet$ from \\eqref{eq:scotty} satisfies the following properties~{\\rm (i)--(iv)} with a constant $C_{\\rm sz}>0$ depending only on $\\gamma,p, \\widehat {\\mathcal{Q}}_0, w_{\\min}, w_{\\max}$, and $\\sigma$:\n\\begin{enumerate}[{\\rm(i)}]\n\\item Local projection property: For all $v\\in L^2(\\Gamma)$ and all $Q\\in{\\mathcal{Q}}_\\bullet$, it holds that \n\\begin{align}\\label{eq:local projection}\n(J_\\bullet v)|_Q= v|_Q \\quad\\text{if }v|_{\\pi_\\bullet^p(Q)}\\in {\\mathcal X}_\\bullet|_{\\pi_\\bullet^p(Q)}=\\set{V_\\bullet|_{\\pi_\\bullet^p(Q)}}{V_\\bullet\\in{\\mathcal X}_\\bullet}.\n\\end{align}\n\\item Local $L^2$-stability: For all $v\\in L^2(\\Gamma)$ and all $Q\\in {\\mathcal{Q}}_\\bullet$, it holds that \n\\begin{align}\\label{eq:local L2}\n\\norm{J_\\bullet v}{L^2(Q)}\\le C_{\\rm sz} \\norm{v}{L^2(\\pi_\\bullet^{p}(Q))}.\n\\end{align}\n\\item Local $H^1$-stability: For all $v\\in H^1(\\Gamma)$ and all $Q\\in {\\mathcal{Q}}_\\bullet$, it holds that \n\\begin{align}\\label{eq:local H1}\n|J_\\bullet v|_{H^1(Q)}\\le C_{\\rm sz} |v|_{H^1(\\pi_\\bullet^{p}(Q))}.\n\\end{align}\n\\item Approximation properties: For all $0\\le\\sigma\\le1$ and all $v\\in H^\\sigma(\\Gamma)$, it holds that\n\\begin{align}\\label{eq:local approx}\n\\norm{h_\\bullet^{-\\sigma}(1-J_\\bullet) v}{L^2(\\Gamma)}\\le C_{\\rm sz} \\norm{v}{H^\\sigma(\\Gamma)}\n\\end{align}\nas well as\n\\begin{align}\\label{eq:local approx2}\n\\norm{(1-J_\\bullet)v}{H^{\\sigma}(\\Gamma)} \\le C_{\\rm sz} \\norm{h_\\bullet^{1-\\sigma}\\partial_\\Gamma v}{L^2(\\Gamma).}\n\\end{align}\n \n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\n\n\\noindent\n\\textbf{Proof of{\\rm \\textbf{(i)}}:}\nThe proof follows immediately from \\eqref{eq:duality} and the fact that\n\\eqref{eq:hypsing basis} forms a basis of ${\\mathcal X}_\\bullet$.\n\n\\noindent\n\\textbf{Proof of {\\rm \\textbf{(ii)}}:}\nAbbreviate $\\overline R_{\\bullet,1-p,p}:=R_{\\bullet,1-p,p}+R_{\\bullet,N_\\bullet-p,p}$.\nBecause of \\eqref{eq:dual inequality} and ${\\rm supp} R_{\\bullet,1-p,p}\\cap{\\rm supp} R_{\\bullet,N_\\bullet-p,p}=\\emptyset$, it holds that \n\\begin{align*}\n\\norm{J_\\bullet v}{L^2(Q)}&\\stackrel{\\eqref{eq:scotty}}{=}\\Big\\|\\Big(\\int_a^b \\frac{\\widehat R_{\\bullet,1-p,p}^*+\\widehat R_{\\bullet,N_\\bullet-p,p}^*}{2} v\\circ\\gamma \\,dt \\Big)\\overline R_{\\bullet,1-p,p}\\\\\n&\\hspace{5cm}+\\sum_{i=1-p}^{N_\\bullet-p-1} \\Big(\\int_a^b \\widehat R_{\\bullet,i,p}^* v\\circ\\gamma \\,dt \\Big)R_{\\bullet,i,p}\\Big\\|_{L^2(Q)}\\\\\n&\\,\\,\\le\\,\\, \\int_a^b \\Big| \\frac{\\widehat R_{\\bullet,1-p,p}^*+\\widehat R_{\\bullet,N_\\bullet-p,p}^*}{2} v\\circ\\gamma\\Big| \\,dt \\,|Q\\cap {\\rm supp}\\overline R_{\\bullet,1-p,p}|^{1\/2}\\\\\n&\\quad+\\sum_{i=1-p\\atop Q\\subseteq {\\rm supp} R_{\\bullet,i,p}}^{N_\\bullet-p-1} \\int_a^b |\\widehat R_{\\bullet,i,p}^* v\\circ\\gamma |\\,dt \\,|Q|^{1\/2}\\\\\n&\\stackrel{\\eqref{eq:dual inequality}}{\\lesssim}|{\\rm supp}\\overline R_{\\bullet,1-p,p}|^{-1\/2}\\norm{v\\circ\\gamma}{L^2({\\rm supp} \\widehat R_{\\bullet,1-p,p}\\cup{\\rm supp} \\widehat R_{\\bullet,N_\\bullet-p,p})}\\,|Q\\cap{\\rm supp} \\overline R_{\\bullet,1-p,p}|^{1\/2}\\\\\n&\\quad+ \\sum_{i=1-p\\atop Q\\subseteq {\\rm supp} R_{\\bullet,i,p}}^{N_\\bullet-p-1} |{\\rm supp} {R}_{\\bullet,i,p}|^{-1\/2}\\norm{v\\circ\\gamma}{L^2({\\rm supp} \\widehat R_{\\bullet,i,p})}|Q|^{1\/2}\\lesssim \\norm{ v}{L^2(\\pi_\\bullet^{p}(Q))}.\n\\end{align*}\n\n\\noindent\n\n\n\\noindent\n\\textbf{Proof of {\\rm \\textbf{(iii)}}:}\nWe show that $|(1-J_\\bullet)v|_{H^1(Q)}\\lesssim |v|_{H^1(\\pi_\\bullet^p(Q))}$. \nWith Lemma~\\ref{lem:properties for B-splines} \\eqref{item:derivative of splines}, for $i=1-p,\\dots,N_\\bullet-p$, it holds that \n\\begin{align*}\n|R_{\\bullet,i,p}|_{H^1(\\Gamma)}&\\simeq |\\widehat R_{\\bullet,i,p}|_{H^1(a,b)}=\\norm{(w_{\\bullet,i}\\widehat B_{\\bullet,i,p}\/\\widehat w)'}{L^2(a,b)}\\simeq\\Big\\|\\frac{\\widehat B_{\\bullet,i,p}'\\widehat w-\\widehat B_{\\bullet,i,p}\\widehat w'}{\\widehat w^2}\\Big\\|_{L^2(a,b)}\\\\\n&\\lesssim\\norm{\\widehat B_{\\bullet,i,p}'}{L^2(a,b)}+\\norm{\\widehat w'}{L^2({\\rm supp} \\widehat B_{\\bullet,i,p})}\\stackrel{\\eqref{item:derivative of splines}}\\lesssim|{\\rm supp} \\widehat B_{\\bullet,i,p}|^{-1\/2}\\simeq |{\\rm supp} {R}_{\\bullet,i,p}|^{-1\/2}.\n\\end{align*}\nWith \\eqref{eq:dual inequality}, we see for $\\widetilde v\\in H^1(\\Gamma)$ that\n\\begin{align}\\label{eqpr:one minus J}\n\\begin{split}\n|J_\\bullet \\widetilde{v}|_{H^1(Q)}&\\stackrel{\\eqref{eq:dual inequality}}\\lesssim |{\\rm supp} \\overline R_{\\bullet,1-p,p}|^{-1\/2} \\norm{\\widetilde{v}\\circ\\gamma}{L^2({\\rm supp} \\widehat R_{\\bullet,1-p,p}\\cup {\\rm supp} \\widehat R_{\\bullet,1-p,p})} |{\\rm supp}\\overline R_{\\bullet,1-p,p}|_{H^1(Q)}\\\\\n&\\quad +\\sum_{i=1-p}^{N_\\bullet-p-1}|{\\rm supp} \\widehat R_{\\bullet,i,p}|^{-1\/2} \\norm{\\widetilde{v}\\circ\\gamma}{L^2({\\rm supp} \\widehat R_{\\bullet,i,p})} |R_{\\bullet,i,p}|_{H^1(Q)}\\\\\n&\\,\\,\\lesssim\\,\\,h_{\\bullet,Q}^{-1\/2} \\norm{\\widetilde{v}}{L^2(\\pi_\\bullet^p(Q))}h_{\\bullet,Q}^{-1\/2}+\\sum_{i=1-p\\atop |Q\\cap {\\rm supp} R_{\\bullet,i,p}|>0}^{N_\\bullet-p-1}h_{\\bullet,Q}^{-1\/2}\\norm{\\widetilde{v}}{L^2(\\pi_\\bullet^p(Q))} h_{\\bullet,Q}^{-1\/2}\\\\\n&\\,\\,\\lesssim\\,\\,\\norm{h_\\bullet^{-1}\\widetilde{v}}{L^2(\\pi_\\bullet^p(Q))}.\n\\end{split}\n\\end{align}\nFor $v\\in H^1(\\Gamma)$, let $\\overline{v}:=\\int_{\\pi_\\bullet^p(Q)} v \\,dx\/|\\pi_\\bullet^p(Q)| $ be the integral mean of $v$ over $\\pi_\\bullet^p(Q)$. \nChoosing $\\widetilde v:=v-\\overline v$ in \\eqref{eqpr:one minus J} and using the Poincar\\'e inequality (see, e.g., \\cite[Lemma 2.5]{faermann2d}), we conclude that \n\\begin{align*}\n|(1-J_\\bullet)v|_{H^1(Q)}&\\,\\,\\,\\,=\\,\\,\\,\\,|(1-J_\\bullet)(v-\\overline v)|_{H^1(Q)}\\\\\n&\\stackrel{\\eqref{eqpr:one minus J}}\\lesssim|v-\\overline v|_{H^1(Q)}+\\norm{h_\\bullet^{-1}(v-\\overline v)}{L^2(\\pi_\\bullet^p(Q))}\\lesssim |v|_{H^1(\\pi_\\bullet^p(Q))}.\n\\end{align*}\n\n\n\n\\noindent\n\\textbf{Proof of {\\rm \\textbf{(iv)}}:}\nFirst, we prove \\eqref{eq:local approx}.\nWith {\\rm (ii)}, it holds that \n\\begin{align*}\n\\norm{(1-J_\\bullet)\\widetilde v}{L^2(Q)}\\lesssim \\norm{\\widetilde v}{L^2(\\pi_\\bullet^p(Q))}\\quad \\text{for all }\\widetilde v\\in L^2(\\Gamma).\n\\end{align*}\nBy taking the square and summing over all elements, this already proves the assertion for $\\sigma=0$.\nNow, we prove it for $\\sigma=1$ by showing that \n\\begin{align}\\label{eq:aux local approx2}\n\\norm{h_\\bullet^{-1}(1-J_\\bullet)v}{L^2(Q)}\\lesssim |{v}|_{H^1(\\pi_\\bullet^p(Q))}\\quad \\text{for all } v\\in L^2(\\Gamma).\n\\end{align}\nWe choose $\\widetilde v:=v-\\overline v$ with $\\overline v:=\\int_{\\pi_\\bullet^p(Q)} v \\,dx\/|\\pi_\\bullet^p(Q)|$ and apply the Poincar\\'e inequality.\nNote, that \\eqref{eq:local approx} for arbitrary $\\sigma$ is equivalent to the boundedness of \n\\begin{align*}\n1-J_\\bullet:(H^\\sigma(\\Gamma),\\norm{\\cdot}{H^\\sigma}(\\Gamma))\\to (L^2(\\Gamma),\\norm{h_\\bullet^{-\\sigma}(\\cdot)}{L^2(\\Gamma)}),\n\\end{align*}\nwhich follows with \\eqref{eq:intp1} and \\eqref{eq:intp2} from the interpolation theorem\n\\cite[Theorem~B.2]{mclean}.\n\n\nNext, we prove \\eqref{eq:local approx}. \nThe localization argument \\cite[Lemma~2.3]{faermann2d} in combination with \\cite[Lemma~4.5]{resigabem} proves that \n$\\norm{\\widetilde v}{H^{1\/2}(\\Gamma)}\n\\lesssim \n\\norm{h_\\bullet^{-1\/2}\\widetilde v}{L^2(\\Gamma)}\n+\\norm{h_\\bullet^{1-1\/2}\\partial_\\Gamma \\widetilde v}{L^2(\\Gamma)}$ for all $\\widetilde v\\in\\widetilde H^{1\/2}(\\Gamma)$.\nThe proofs extend verbatim to $0<\\sigma<1$ and the assertion is trivially satisfied for $\\sigma\\in\\{0,1\\}$. \nWith $\\widetilde v:=(1-J_\\bullet)v$, the latter inequality (for $0\\le \\sigma\\le1$), \\eqref{eq:aux local approx2}, and \\eqref{eq:local H1} show that \n\\begin{eqnarray*}\n\\norm{(1-J_\\bullet)v}{H^\\sigma(\\Gamma)}\n&\\stackrel{\\text{\\cite{faermann2d,resigabem}}}\\lesssim &\n\\norm{h_\\bullet^{-\\sigma}(1-J_\\bullet)v}{L^2(\\Gamma)}\n+\\norm{h_\\bullet^{1-\\sigma}\\partial_\\Gamma (1-J_\\bullet)v}{L^2(\\Gamma)}\n\\\\\n&\\stackrel{\\eqref{eq:aux local approx2}+\\eqref{eq:local H1}}\\lesssim&\n \\norm{h_\\bullet^{1-\\sigma}\\partial_\\Gamma v}{L^2(\\Gamma)}.\n\\end{eqnarray*}\nThis concludes the proof.\n\n\\end{proof}\n\n\n\n\\subsection{Inverse inequalities}\n\\label{sec:inverse inequalities}\n\\label{sec:inverse inequalities}\nThe first result is taken from \\cite[Theorem~3.1]{invest}.\n\\begin{proposition}\nLet ${\\mathcal K}_\\bullet\\in\\mathbb{K}$.\nThen, there exists a constant $C_{\\rm inv}>0$ such that \n\\begin{align}\\label{eq:inverse W}\n\\norm{h_\\bullet^{1\/2}\\mathfrak{W} v}{L^2(\\Gamma)}\\le C_{\\rm inv} \\big(\\norm{v}{H^{1\/2}(\\Gamma)} + \\norm{h_\\bullet^{1\/2}\\partial_\\Gamma v}{L^2(\\Gamma)}\\big)\\quad \\text{for all }v\\in H^1(\\Gamma)\n\\end{align}\nand \n\\begin{align}\\label{eq:inverse Kpr}\n\\norm{h_\\bullet^{1\/2} \\mathfrak{K}'\\psi}{L^2(\\Gamma)}\\le C_{\\rm inv} \\big(\\norm{\\psi}{H^{-1\/2}(\\Gamma)} + \\norm{h_\\bullet^{1\/2}\\psi}{L^2(\\Gamma)}\\big)\\quad\\text{for all }\\psi\\in L^2(\\Gamma).\n\\end{align}\nThe constant $C_{\\rm inv}>0$ depends only on $\\gamma$ and $\\widehat{\\mathcal{Q}}_0$.\n\\hfill$\\square$\n\\end{proposition}\n\n\nThe next proposition provides inverse inequalities for rational splines, which are well-known for piecewise polynomials; see \\cite{inverse2,hypsing3d}.\nIt also recalls a standard inverse inequality for piecewise polynomials. \n\n\n\n\n\n\\begin{proposition}\\label{lem:discrete invest2}\nLet ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ and $0\\le\\sigma\\le 1$.\nThen, there exists $\\widetilde C_{\\rm inv}>0$ such that \n\\begin{align}\\label{eq:discrete invest}\n\\norm{h_\\bullet^{1-\\sigma}\\partial_\\Gamma{V_\\bullet}}{L^2(\\Gamma)}\\le \\widetilde C_{\\rm inv} \\norm{V_\\bullet}{H^\\sigma(\\Gamma)} \\quad\\text{for all } V_\\bullet\\in {\\mathcal X}_\\bullet,\n\\end{align}\nand\n\\begin{align}\\label{eq:discrete invest old}\n\\norm{h_\\bullet^{\\sigma}\\Psi_\\bullet}{L^2(\\Gamma)}\\le\\widetilde C_{\\rm inv}\\norm{\\Psi_\\bullet}{H^{-\\sigma}(\\Gamma)}\\quad\\text{for all }\\Psi_\\bullet\\in\\mathbf{P}^p({\\mathcal{Q}}_\\bullet),\n\\end{align}\nand\n\\begin{align}\\label{eq:discrete invest2}\n\\norm{V_\\bullet}{H^\\sigma(\\Gamma)}\\le \\widetilde C_{\\rm inv} \\norm{h_\\bullet^{-\\sigma}V_\\bullet}{L^2(\\Gamma)} \\quad\\text{for all } V_\\bullet\\in {\\mathcal X}_\\bullet.\n\\end{align}\nThe constant $\\widetilde C_{\\rm inv}>0$ depends only on $\\gamma, p, \\widehat{\\mathcal{Q}}_0, w_{\\min}, w_{\\max}$, and $\\sigma$.\n\\end{proposition}\n\\begin{proof}\n\\eqref{eq:discrete invest old} is proved, e.g., in \\cite[Proposition~4.1]{optigabem} even for piecewise rational splines. \nWe prove the other two assertions in three steps.\n\n\\noindent\n\\textbf{Step 1:} We prove that \\eqref{eq:discrete invest} and \\eqref{eq:discrete invest2} hold (even elementwise) for $\\sigma\\in\\{0,1\\}$.\nWe start with \\eqref{eq:discrete invest}.\nFor $\\sigma=1$ the assertion is trivial. \nIf $\\sigma=0$, let $Q\\in{\\mathcal{Q}}_\\bullet$, define $\\widehat{Q}:=\\gamma^{-1}(Q)$, and let $\\Phi_{\\widehat{Q}}$ be the affine bijection which maps $[0,1]$ onto $\\widehat Q$.\nThen, it holds that\n\\begin{align*}\n|V_\\bullet|_{H^1(Q)}^2&=\\int_Q( \\partial_\\Gamma V_\\bullet)^2 \\,dx=\\int_{\\widehat{Q}} (V_\\bullet\\circ\\gamma)'(t)^2\/|\\gamma'(t)| \\,dt=|\\widehat Q|\\int_0^1 (V_\\bullet\\circ\\gamma)'(\\Phi_{\\widehat{Q}}(t))^2\/|\\gamma'(\\Phi_{\\widehat Q}(t))|\\,dt\\\\\n&\\simeq |\\widehat{Q}|^{-1}\\int_0^1 (V_\\bullet\\circ\\gamma\\circ\\Phi_{\\widehat Q})'(t)^2\\,dt=|\\widehat{Q}|^{-1}|V_\\bullet\\circ\\gamma\\circ\\Phi_{\\widehat Q}|_{H^1(0,1)}^2.\n\\end{align*}\nNote that $V_\\bullet\\circ\\gamma\\circ\\Phi_{\\widehat{Q}}$ is just a rational function on the interval $[0,1]$. \nIt can be written as $q\/\\widetilde w$ with some polynomials $q,\\widetilde w\\in \\mathcal{P}^p(0,1)$ of degree $p$, where $00$ depends only on $p$, $w_{\\min}$, $w_{\\max}$. \nWith the quotient rule, we conclude that\n\\begin{align*}\n|V_\\bullet\\circ\\gamma\\circ\\Phi_{\\widehat{Q}}|_{H^1(0,1)}=|q\/\\widetilde{w}|_{H^1(0,1)}\\lesssim \\norm{q}{L^2(0,1)}\\simeq\\norm{V_\\bullet\\circ\\gamma\\circ\\Phi_{\\widehat{Q}}}{L^2(0,1)}.\n\\end{align*}\nThis shows that \n\\begin{align*}\n|V_\\bullet|^2_{H^1(Q)}\\lesssim|\\widehat{Q}|^{-1} \\norm{V_\\bullet\\circ\\gamma\\circ\\Phi_{\\widehat Q}}{L^2(0,1)}^2\\simeq \\norm{h_\\bullet^{-1} V_\\bullet}{L^2(Q)}^2.\n\\end{align*}\nNow, we consider \\eqref{eq:discrete invest2}.\nFor $\\sigma=0$, the assertion is trivial.\nThe case $\\sigma=1$ follows from \\eqref{eq:discrete invest} with $\\sigma=0$.\n\n\n\\noindent\n\\textbf{Step 2:}\nWe prove \\eqref{eq:intp1} and \\eqref{eq:intp2} for the discrete space ${\\mathcal X}_\\bullet$. Note that Proposition \\ref{lem:Scott properties} proves that $J_\\bullet$ is a stable projection onto ${\\mathcal X}_\\bullet$ considered as a mapping from $(L^2(\\Gamma),\\norm{\\cdot}{L^2(\\Gamma)})$ to $({\\mathcal X}_\\bullet,\\norm{\\cdot}{L^2(\\Gamma)})$, from $(L^2(\\Gamma),\\norm{h^{-1}(\\cdot)}{L^2(\\Gamma)})$ to $({\\mathcal X}_\\bullet,\\norm{h^{-1}(\\cdot)}{L^2(\\Gamma)})$, or from $(H^1(\\Gamma),\\norm{\\cdot}{H^1(\\Gamma)})$ to $({\\mathcal X}_\\bullet,\\norm{\\cdot}{H^1(\\Gamma)})$.\nDue to \\eqref{eq:intp1} and \\eqref{eq:intp2}, Lemma~\\ref{lem:discrete interpolation} is applicable and proves that\n\\begin{align}\\label{eq:intp3}\n({\\mathcal X}_\\bullet,\\norm{h_\\bullet^{-\\sigma}{(\\cdot})}{L^2(\\Gamma)})\\simeq [({\\mathcal X}_\\bullet,\\norm{\\cdot}{L^2(\\Gamma)}),({\\mathcal X}_\\bullet,\\norm{h_\\bullet^{-1}(\\cdot)}{L^2(\\Gamma)})]_\\sigma,\n\\end{align}\nand \n\\begin{align}\\label{eq:intp4}\n({\\mathcal X}_\\bullet,\\norm{\\cdot}{H^\\sigma(\\Gamma)})\\simeq[({\\mathcal X}_\\bullet,\\norm{\\cdot}{L^2(\\Gamma)}), ({\\mathcal X}_\\bullet,\\norm{\\cdot}{H^1(\\Gamma)})]_\\sigma.\n\\end{align}\n\n\n\\noindent\n\\textbf{Step 3:}\nConsider the differentiation operator\n\\begin{align*}\n{ \\partial_\\Gamma}:({\\mathcal X}_\\bullet,\\norm{\\cdot}{H^\\sigma(\\Gamma)})\\to (L^2(\\Gamma),\\norm{h_\\bullet^{1-\\sigma}(\\cdot)}{L^2(\\Gamma)}),\n\\end{align*}\nand the formal identity\n\\begin{align*}\n{\\rm id}:({\\mathcal X}_\\bullet,\\norm{h_\\bullet^{-\\sigma}(\\cdot)}{L^2(\\Gamma)})\\to(H^\\sigma(\\Gamma),\\norm{\\cdot}{H^\\sigma(\\Gamma)}).\n\\end{align*}\nThen, \\eqref{eq:discrete invest} resp. \\eqref{eq:discrete invest2} is equivalent to boundedness of $\\partial_\\Gamma$ resp. ${\\rm id}$.\nFor $\\sigma\\in\\{0,1\\}$, $\\partial_\\Gamma$ and ${\\rm id}$ are bounded according to Step 1. \nFinally, Step 2 and the well-known interpolation theorem \\cite[Theorem B.2]{mclean} prove boundedness of the mappings $\\partial_\\Gamma$ and ${\\rm id}$.\n\\end{proof}\n\n\n\\subsection{Stability on non-refined node patches (E1)}\n\\label{sec:hypsing stability}\nNote that $\\pi_\\bullet(z)=\\pi_\\circ(z)$ if $z\\in{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}$ and that $\\widetilde h_{\\bullet,z}=\\widetilde h_{\\circ,z}$ for $z\\in{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}$, which follows from the definitions \\eqref{eq:tilde h equivalent} and \\eqref{eq:nnr}.\nThe reverse triangle inequality proves that\n\\begin{align*}\n&|\\widetilde\\eta_\\circ({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})-\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})|\\\\\n&\\quad\\le \\Big| \\sum_{ z\\in{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}}\\big(\\norm{\\widetilde h_{\\circ, z}^{1\/2} (g_\\circ-\\mathfrak{W}U_\\circ)}{L^2(\\pi_\\circ(z))} -\\norm{\\widetilde h_{\\bullet, z}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(\\pi_\\circ(z))} \\big)^2\\Big|^{1\/2}\\\\\n&\\qquad+ \\Big| \\sum_{ z\\in{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}}\\big(\\norm{\\widetilde h_{\\circ, z}^{1\/2} (\\phi-\\phi_\\circ)}{L^2(\\pi_\\circ(z))} -\\norm{\\widetilde h_{\\bullet, z}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(\\pi_\\circ(z))} \\big)^2\\Big|^{1\/2}\\\\\n&\\quad\\le \\Big| \\sum_{ z\\in{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}}\\norm{\\widetilde h_{\\circ, z}^{1\/2} \\mathfrak{W}(U_\\circ-U_\\bullet)}{L^2(\\pi_\\circ(z))} ^2\\Big|^{1\/2}\n+\\Big| \\sum_{ z\\in{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}}\\norm{\\widetilde h_{\\circ, z}^{1\/2} (g_\\circ-g_\\bullet)}{L^2(\\pi_\\circ(z))} ^2\\Big|^{1\/2}\\\\\n&\\qquad+\\Big| \\sum_{ z\\in{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}}\\norm{\\widetilde h_{\\circ, z}^{1\/2} (\\phi_\\circ-\\phi_\\bullet)}{L^2(\\pi_\\circ(z))} ^2\\Big|^{1\/2}.\n\\end{align*}\nThe regularity of $\\gamma$, local quasi-uniformity \\eqref{eq:two kappa}, and the equivalence \\eqref{eq:tilde h equivalent} yield that \n\\begin{align*}\n|\\widetilde\\eta_\\circ({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})-\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})|\n\\lesssim\\norm{h_\\circ^{1\/2} \\mathfrak{W}(U_\\circ-U_\\bullet)}{L^2(\\Gamma)}+\\norm{h_\\circ^{1\/2} (g_\\circ-g_\\bullet)}{L^2(\\Gamma)}+\\norm{h_\\circ^{1\/2} (\\phi_\\circ-\\phi_\\bullet)}{L^2(\\Gamma)}.\n\\end{align*}\nIf ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$, nestedness \\eqref{eq:nested} shows that $U_\\circ-U_\\bullet\\in{\\mathcal X}_\\circ$. \nOtherwise, we define ${\\mathcal K}_{\\ell\\cup(\\ell+1)}\\in\\mathbb{K}$ via ${\\mathcal N}_{\\ell\\cup(\\ell+1)}:={\\mathcal N}_\\ell\\cup{\\mathcal N}_{\\ell+1}={\\mathcal N}_{\\ell+1}$ and $\\#_{\\ell\\cup(\\ell+1)} z:=\\max\\{\\#_\\ell z,\\#_{\\ell+1} z\\}$ for all $z\\in{\\mathcal N}_{\\ell+1}$, where $\\#_\\ell z:=0$ if $z\\not\\in{\\mathcal N}_\\ell$.\nThen, $U_\\circ-U_\\bullet\\in{\\mathcal X}_{\\ell\\cup(\\ell+1)}$ and $h_{\\ell\\cup(\\ell+1)}=h_\\circ$.\nTherefore, in each case, the inverse inequalities \\eqref{eq:inverse W}--\\eqref{eq:discrete invest old} are applicable and conclude the proof.\nThe overall constant $\\const{stab}$ depends only on the parametrization $\\gamma$, the polynomial order $p$, and the initial mesh $\\widehat{\\mathcal{Q}}_0$.\n\n\n\\subsection{Reduction on refined node patches (E2)}\nLet $\\delta>0$.\nWe apply the triangle inequality and the Young inequality to see that\n\\begin{align*}\n&\\widetilde\\eta_\\circ({\\mathcal N}^{\\rm ref}_{\\circ,\\bullet})^2=\\sum_{ z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}} \\Big(\\norm{\\widetilde h_{\\circ,z}^{1\/2} (g_\\circ-\\mathfrak{W}U_\\circ)}{L^2(\\pi_\\circ(z))}^2+\\norm{\\widetilde h_{\\circ,z}^{1\/2} (\\phi-\\phi_\\circ)}{L^2(\\pi_\\circ(z))}^2\\Big)\\\\\n&\\le \\sum_{ z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}}\\Big( (1+\\delta)^2\\norm{\\widetilde h_{\\circ,z}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(\\pi_\\circ(z))}^2+(1+\\delta^{-1})\\norm{\\widetilde h_{\\circ,z}^{1\/2}\\mathfrak{W}(U_\\circ-U_\\bullet)}{L^2(\\pi_\\circ(z))}^2\\\\\n&\\hspace{9cm}+(1+\\delta)(1+\\delta^{-1})\\norm{\\widetilde h_{\\circ,z}^{1\/2}(g_\\circ-g_\\bullet)}{L^2(\\pi_\\circ(z))}^2\\Big)\\\\\n&\\quad+ \\sum_{ z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}} \\Big((1+\\delta)\\norm{\\widetilde h_{\\circ,z}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(\\pi_\\circ(z))}^2+(1+\\delta^{-1})\\norm{\\widetilde h_{\\circ,z}^{1\/2}(\\phi_\\circ-\\phi_\\bullet)}{L^2(\\pi_\\circ(z))}^2\\Big).\n\\end{align*}\nWe only have to estimate the first terms in each of the last two sums, the other terms can be estimated as in Section~\\ref{sec:hypsing stability}.\nWe split each patch $\\pi_\\circ(z)=Q_{\\circ,{\\rm left}}(z)\\cup Q_{\\circ,{\\rm right}}(z)$ into a (with respect to the parametrization $\\gamma$) left and a right element in ${\\mathcal{Q}}_\\circ$.\nWe obtain that\n\\begin{align*}\n&\\sum_{ z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}} \\Big(\\norm{\\widetilde h_{\\circ,z}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(\\pi_\\circ(z))}^2+ \\norm{\\widetilde h_{\\circ,z}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(\\pi_\\circ(z))}^2\\Big)\\\\\n&\\quad=\n\\sum_{ z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}}\\Big( \\norm{\\widetilde h_{\\circ,z}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(Q_{\\circ,{\\rm left}}(z))}^2+ \\norm{\\widetilde h_{\\circ,z}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(Q_{\\circ,{\\rm left}}(z))}^2\\Big)\n\\\\&\\qquad+\\sum_{ z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}} \\Big(\\norm{\\widetilde h_{\\circ,z}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(Q_{\\circ,{\\rm right}}(z))}^2+\\norm{\\widetilde h_{\\circ,z}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(Q_{\\circ,{\\rm right}}(z))}^2\\Big).\n\\end{align*}\nLet $z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}$.\nIf $z\\in{\\mathcal N}_\\bullet$, we define $z':=z$ and note that $z'\\in {\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}$. \n Otherwise, there exists a unique $z'\\in{\\mathcal N}_\\bullet$ with $z\\in Q_{\\bullet,{\\rm left}}(z')$, where $Q_{\\bullet,{\\rm left}}(z')$ is defined analogously as above.\nAgain, this implies that $z'\\in{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}$.\nAltogether, the contraction property~\\eqref{eq:h tilde ctr} yields that \n\\begin{align*}\n&\\sum_{ z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}} \\Big(\\norm{\\widetilde h_{\\circ,z}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(Q_{\\circ,{\\rm left}}(z))}^2+ \\norm{\\widetilde h_{\\circ,z}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(Q_{\\circ,{\\rm left}}(z))}^2\\Big)\\\\\n&\\quad\\,\\,\\le\\,\\,\\sum_{ z'\\in{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}}\n\\sum_{\\quad{z=z' \\text{ or } z\\in Q_{\\bullet,{\\rm left}}(z')\\setminus{\\mathcal N}_\\bullet}} \n\\Big(\n\\norm{\\widetilde h_{\\circ,z}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(Q_{\\circ,{\\rm left}}(z))}^2\n+\\norm{\\widetilde h_{\\circ,z}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(Q_{\\circ,{\\rm left}}(z))}^2\\Big)\\\\\n&\\quad\\stackrel{\\eqref{eq:h tilde ctr}}\\le \\sum_{ z'\\in{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ} }\\ro{ctr} \\Big(\\,\\norm{\\widetilde h_{\\bullet,z'}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(Q_{\\bullet,{\\rm left}}(z'))}^2+\\norm{\\widetilde h_{\\bullet,z'}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(Q_{\\bullet,{\\rm left}}(z'))}^2\\Big).\n\\end{align*}\nThe same holds for the right elements. \nHence, we end up with \n\\begin{align*}\n&\\sum_{ z\\in{\\mathcal N}^{\\rm ref}_{\\circ,\\bullet}} \\norm{\\widetilde h_{\\circ,z}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(\\pi_{\\circ}(z))}^2+ \\norm{\\widetilde h_{\\circ,z}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(\\pi_{\\circ}(z))}^2\\\\\n&\\le \\ro{ctr} \\Big(\\sum_{z'\\in{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ} }\\norm{\\widetilde h_{\\bullet,z'}^{1\/2} (g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(\\pi_{\\bullet}(z'))}^2+\\norm{\\widetilde h_{\\bullet,z'}^{1\/2} (\\phi-\\phi_\\bullet)}{L^2(\\pi_{\\bullet}(z'))}^2\\Big)=\\ro{ctr}\\,\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ})^2.\n\\end{align*}\nChoosing $\\delta$ sufficiently small such that $\\ro{red}:=(1+\\delta)^2 \\ro{ctr}<1$, we conclude the proof. \nMoreover, our argument shows that $\\const{red}\\simeq(1+\\delta^{-1})\\const{stab}^2$ with a generic hidden constant.\n\n\n\n\n\n\n\n\n\n\\subsection{Discrete reliability (E3)}\n\\label{sec:hypsing discrete reliability}\nWe show that there exist constants $\\const{drel},\\const{ref}\\ge 1$ such that for all ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ and all ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$, the subset \n\\begin{align}\\label{eq:RR defined2}\n\\RR_{\\bullet,\\circ}:={\\mathcal N}_\\bullet\\cap\\pi_\\bullet^{2p+1}({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}\n\\end{align}\n satisfies that\n\\begin{align}\\label{eq:drel start}\n\\big(\\norm{U_\\circ-U_\\bullet}{H^{1\/2}(\\Gamma)}^2+\\norm{\\phi_\\circ-\\phi_\\bullet}{H^{-1\/2}(\\Gamma)}^2\\big)^{1\/2}\\le \\const{drel}\\,\\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ}),\n\\end{align}\nwith\n\\begin{align}\\label{eq:aux1}\n{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}\\subseteq\\RR_{\\bullet,\\circ}\\quad\n\\text{and}\\quad\n\\#_\\bullet\\RR_{\\bullet,\\circ}\\le\\const{ref} (\\#_\\circ {\\mathcal N}_\\circ-\\#_\\bullet{\\mathcal N}_\\bullet).\n\\end{align}\nObviously, ${\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}\\subseteq\\RR_{\\bullet,\\circ}$ is satisfied. \nHence, the first property of \\eqref{eq:aux1}, i.e., is obvious.\nSince the maximal knot multiplicity is bounded by $p+1$, it holds that \n\\begin{align*}\n\\#_\\bullet\\RR_{\\bullet,\\circ}\\le(p+1) |\\RR_{\\bullet,\\circ}|\\le(p+1)(4p+3)|{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}|\\simeq \\#_\\bullet{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}, \n\\end{align*}\nwhere the hidden constant depends only on $p$.\nNote that $z\\in{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}$ holds only if a knot is inserted in the corresponding patch $\\pi_\\bullet(z)$, where a new knot can be inserted in at most three old patches.\nSince $\\#_\\circ {\\mathcal N}_\\circ-\\#_\\bullet{\\mathcal N}_\\bullet$ is the number of all new knots, we see that\n\\begin{align*}\n\\#_\\bullet{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}\\le 3\\,(\\#_\\circ {\\mathcal N}_\\circ-\\#_\\bullet{\\mathcal N}_\\bullet).\n\\end{align*}\nIn the following four steps, we \nprove \\eqref{eq:drel start}.\n\n\\noindent\n\\textbf{Step 1:} \nLet $U_{\\circ,\\bullet}$ denote the unique Galerkin solution to\n\\begin{align}\n\\edual{U_{\\circ,\\bullet}}{ V_\\circ} =\\dual{g_\\bullet}{V_\\circ}_{L^2(\\Gamma)}\\quad\\text{for all }V_\\circ \\in{\\mathcal X}_\\circ.\n\\end{align}\nEllipticity and the definition of $U_\\circ$ as well as $U_{\\circ,\\bullet}$ show that \n\\begin{align*}\n\\norm{U_\\circ-U_{\\circ,\\bullet}}{H^{1\/2}(\\Gamma)}^2 &\\lesssim \\edual{U_\\circ-U_{\\circ,\\bullet}}{U_\\circ-U_{\\circ,\\bullet}}= \\dual{g_\\circ-g_\\bullet}{U_\\circ-U_{\\circ,\\bullet}}_{L^2(\\Gamma)}\\\\\n&\\le \\norm{g_\\circ-g_\\bullet}{H^{-1\/2}(\\Gamma)}\\norm{U_\\circ-U_{\\circ,\\bullet}}{H^{1\/2}(\\Gamma)}.\n\\end{align*}\nTogether with continuity of $\\mathfrak{K}'$, this yields that \n\\begin{align}\\label{eq:ufine minus uqfine}\n\\norm{U_\\circ-U_{\\circ,\\bullet}}{H^{1\/2}(\\Gamma)} \\lesssim \\norm{g_\\circ-g_\\bullet}{H^{-1\/2}(\\Gamma)}\\lesssim \\norm{\\phi_\\circ-\\phi_\\bullet}{H^{-1\/2}(\\Gamma)}.\n\\end{align}\nMoreover, the triangle inequality and the Young inequality prove that \n\\begin{align}\n&\\big(\\norm{U_\\circ-U_\\bullet}{H^{-1\/2}(\\Gamma)}+\\norm{\\phi_\\circ-\\phi_\\bullet}{H^{-1\/2}(\\Gamma)}\\big)^2\\notag\\\\\n&\\quad\\,\\,\\,\\lesssim\\,\\,\\, \\norm{U_\\circ-U_{\\circ,\\bullet}}{H^{1\/2}(\\Gamma)}^2 + \\norm{U_{\\circ,\\bullet}-U_\\bullet}{H^{1\/2}(\\Gamma)}^2 + \\norm{\\phi_\\circ-\\phi_\\bullet}{H^{-1\/2}(\\Gamma)}^2\\notag\\\\\n&\\quad\\stackrel{\\eqref{eq:ufine minus uqfine}}\\lesssim \\norm{U_{\\circ,\\bullet}-U_\\bullet}{H^{1\/2}(\\Gamma)} +\\norm{\\phi_\\circ-\\phi_\\bullet}{H^{-1\/2}(\\Gamma)}. \\label{eq:drel 3}\n\\end{align}\n\n\n\n\\noindent\n\\textbf{Step 2:}\nWe estimate the last term in \\eqref{eq:drel 3}. \nSince the orthogonal projections $P_\\bullet,P_\\circ$ onto the space of (transformed) piecewise polynomials satisfy that $P_\\circ(1-P_\\bullet)=P_\\circ-P_\\bullet=(1-P_\\bullet) P_\\circ$, the approximation property \\cite[Theorem~4.1]{cp06} shows that \n\\begin{align}\\label{eq:proj proj}\n\\begin{split}\n\\norm{\\phi_\\circ-\\phi_\\bullet}{H^{-1\/2}(\\Gamma)} \n=\\norm{(P_{\\circ}-P_\\bullet)\\phi}{H^{-1\/2}(\\Gamma)}\n&\\stackrel{\\text{\\cite{cp06}}}\\lesssim \\norm{h_\\bullet^{1\/2}(P_{\\circ}-P_\\bullet)\\phi}{L^2(\\Gamma)}\\\\\n&\\,\\,\\,\\,\\lesssim \\,\\,\\,\\,\\norm{h_\\bullet^{1\/2}(\\phi- \\phi_\\bullet)}{L^2(\\bigcup ({\\mathcal{Q}}_\\bullet\\setminus{\\mathcal{Q}}_\\circ))}.\n\\end{split}\n\\end{align}\nNote that $\\bigcup ({\\mathcal{Q}}_\\bullet\\setminus{\\mathcal{Q}}_\\circ)\\subseteq\\pi_\\bullet(\\RR_{\\bullet,\\circ})$. \nTogether with the equivalence \\eqref{eq:tilde h equivalent}, we obtain that \n\\begin{align}\n\\norm{\\phi_\\circ-\\phi_\\bullet}{H^{-1\/2}(\\Gamma)}\\lesssim \\widetilde{\\rm osc}_\\bullet(\\RR_{\\bullet,\\circ}).\n\\end{align}\n\n\\noindent\n\\textbf{Step 3:} \nTo proceed, we apply the projection property \\eqref{eq:local projection} for $V_\\circ:=U_{\\circ,\\bullet}-U_\\bullet$.\nLet $Q\\in{\\mathcal{Q}}_\\bullet\\setminus \\Pi_\\bullet^{2p+1}({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ})$.\nWe show that \n\\begin{align}\\label{eq:S local bem2}\nV_\\circ|_{\\pi_\\bullet^p(Q)}\\in {\\mathcal X}_\\bullet|_{\\pi_\\bullet^p(Q)}=\\set{V_\\bullet|_{\\pi_\\bullet^p(Q)}}{V_\\bullet\\in{\\mathcal X}_\\bullet}, \n\\end{align}\nwherefore \\eqref{eq:local projection} will imply that \n\\begin{align}\\label{eq:S local bem2 imply}\n(1-J_\\bullet)(U_{\\circ,\\bullet}-U_\\bullet)=0\\quad\\text{on}\\quad\\Gamma\\setminus \\pi_\\bullet^{2p+1}({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}).\n\\end{align}\nFirst, we argue by contradiction to see that\n\\begin{align}\\label{eq:inverse patch bem2}\n\\Pi_\\bullet^{2p}(Q)\\subseteq \\Pi_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ}).\n\\end{align}\nSuppose there exists $Q'\\in\\Pi_\\bullet^{2p}(Q)$ with $Q'\\not\\in\\Pi_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})$.\nThis is equivalent to $Q\\in\\Pi_\\bullet^{2p}(Q')$ and $Q'\\in {\\mathcal{Q}}_\\bullet\\setminus\\Pi_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})$, which yields that $Q\\in\\Pi_\\bullet^{2p}\\big({\\mathcal{Q}}_\\bullet\\setminus\n\\Pi_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ}))$.\nNote that \n\\begin{align*}\n{\\mathcal{Q}}_\\bullet\\setminus\\Pi_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})\\subseteq\\Pi_\\bullet({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}),\n\\end{align*}\nsince $Q''$ in the left-hand side implies that $Q''\\cap{\\mathcal N}_\\bullet\\neq\\emptyset$ and $z\\not\\in Q''$ for all $z\\in{\\mathcal N}^{\\rm id}_{\\bullet,\\circ}$, and hence implies the existence of $z\\in{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}$ with $z\\in Q''$.\nAltogether, we see that\n\\begin{align*}\nQ\\in\\Pi_\\bullet^{2p+1}({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}),\n\\end{align*}\nwhich contradicts that $Q\\in{\\mathcal{Q}}_\\bullet\\setminus \\Pi_\\bullet^{2p+1}({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ})$ and thus proves \\eqref{eq:inverse patch bem2}.\n\nNext, we prove \\eqref{eq:S local bem2}.\nNote that $V_\\circ|_{\\pi_\\bullet(Q)}$ can be written as linear combination of (transformed) B-splines $B_{\\circ,i,p}:=\\widehat B_{\\circ,i,p}\\circ\\gamma^{-1}$ that have support on $\\pi_\\bullet(Q)$.\nBy Lemma~\\ref{lem:properties for B-splines} \\eqref{item:B-splines local}, ${\\rm supp}(B_{\\circ,i,p})$ is connected and consists of at most $p+1$ elements, which implies that ${\\rm supp}(B_{\\circ,i,p})\\subseteq\\pi_\\bullet^{2p}(T)$.\nWe show that no knots are inserted in $\\pi_\\bullet^{2p}(Q)$ \nand thus in ${\\rm supp}(B_{\\circ,i,p})$ \nduring the refinement from ${\\mathcal K}_\\bullet$ to ${\\mathcal K}_\\circ$.\nTo see this, let $z'\\in{\\mathcal N}_\\circ$ be a corresponding node.\nSince ${\\mathcal N}^{\\rm id}_{\\bullet,\\circ}$ is just the set of all nodes $z$ such that no new knot is inserted in the patch of $z$, $z'$ cannot belong to $\\pi_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})$.\nHence, \\eqref{eq:inverse patch bem2} implies that $z'\\not\\in\\pi_\\bullet^{2p}(Q)$.\nWith this, Lemma~\\ref{lem:properties for B-splines} \\eqref{item:B-splines determined} proves that $B_{\\circ,i,p}=B_{\\bullet,i',p}$ for some B-spline $B_{\\bullet,i',p}:=\\widehat B_{\\circ,i',p}\\circ\\gamma^{-1}$.\nIn particular, $V_\\circ|_{\\pi_\\bullet(Q)}$ can be written as linear combination of (transformed) B-splines corresponding to ${\\mathcal K}_\\bullet$, which implies \\eqref{eq:S local bem2} and \\eqref{eq:S local bem2 imply}.\n\n\n\\noindent\n\\textbf{Step 4:} \nIt remains to estimate the second term in \\eqref{eq:drel 3}.\nDue to ellipticity as well as Galerkin orthogonality, we see that \n\\begin{align*}\n\\norm{U_{\\circ,\\bullet}-U_\\bullet}{H^{1\/2}(\\Gamma)}^2&\\lesssim \\edual{ U_{\\circ,\\bullet}-U_\\bullet}{U_{\\circ,\\bullet}-U_\\bullet} =\\edual{ U_{\\circ,\\bullet}-U_\\bullet}{(1-J_\\bullet)(U_{\\circ,\\bullet}-U_\\bullet)}.\n\\end{align*}\nIt holds that\n\\begin{align*}\n0=\\edual{ U_{\\circ,\\bullet}-U_\\bullet}{1}= \\dual{\\mathfrak{W} (U_{\\circ,\\bullet}-U_\\bullet)}{1}_{L^2(\\Gamma)} +\\dual{U_{\\circ,\\bullet}-U_\\bullet}{1}_{L^2(\\Gamma)} \\dual{1}{1}_{L^2(\\Gamma)}.\n\\end{align*}\nSince $ \\dual{\\mathfrak{W} (U_{\\circ,\\bullet}-U_\\bullet)}{1}_{L^2(\\Gamma)}=0$, $U_{\\circ,\\bullet}-U_\\bullet$ has integral mean zero.\nAltogether, we see that \n\\begin{align}\\label{eq:star plus minus star}\n\\norm{U_{\\bullet,\\circ}-U_\\bullet}{H^{1\/2}(\\Gamma)}^2\\lesssim\n\\dual{g_\\bullet-\\mathfrak{W} U_\\bullet}{(1-J_\\bullet)(U_{\\circ,\\bullet}-U_\\bullet)}_{L^2(\\Gamma)}.\n\\end{align}\nWith \\eqref{eq:S local bem2 imply} of Step~3 and the Cauchy-Schwarz inequality, we thus obtain that\n\\begin{align*}\n&\\norm{U_{\\circ,\\bullet}-U_\\bullet}{H^{1\/2}(\\Gamma)}^2\\stackrel{\\eqref{eq:S local bem2 imply}}{\\lesssim} \\dual{ g_{\\bullet}-\\mathfrak{W}U_\\bullet}{(1-J_\\bullet)(U_{\\circ,\\bullet}-U_\\bullet)}_{L^2(\\pi_\\bullet^{2p+1}({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}))}\\\\\n&\\le \\norm{ h_\\bullet^{1\/2}(g_\\bullet-\\mathfrak{W}U_\\bullet)}{L^2(\\pi_\\bullet^{2p+1}({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}))} \n\\norm{ h_\\bullet^{-1\/2}((1-J_\\bullet)(U_{\\circ,\\bullet}-U_\\bullet)}{L^2(\\pi_\\bullet^{2p+1}({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}))}.\n\\end{align*}\nThe equivalence \\eqref{eq:tilde h equivalent} and the approximation property \\eqref{eq:local approx} yield that\n\\begin{align*}\n\\norm{U_{\\circ,\\bullet}-U_\\bullet}{H^{1\/2}(\\Gamma)}^2\\lesssim \\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ})\\norm{U_{\\circ,\\bullet}-U_\\bullet}{H^{1\/2}(\\Gamma)}.\n\\end{align*}\nThis concludes the proof.\nThe constants $\\const{drel},\\const{ref}$ depend only on the parametrization $\\gamma$, the polynomial order $p$, and the initial mesh $\\widehat{\\mathcal{Q}}_0$.\n\n\\subsection{Reliability in (\\ref{eq:reliable})}\n\\label{sec:reliable hyper}\nWe only consider the case $\\phi_\\bullet:=\\phi$ for all ${\\mathcal K}_\\bullet\\in\\mathbb{K}$.\nThe other case, i.e., $\\phi_\\bullet:=P_\\bullet\\phi$ for all ${\\mathcal K}_\\bullet\\in\\mathbb{K}$, follows analogously.\n\n\\noindent\n\\textbf{Step 1:} First, we show that for arbitrary $\\varepsilon>0$, there exists a refinement ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$ such that $\\norm{u-U_\\circ}{H^{1\/2}(\\Gamma)}\\le\\varepsilon$.\nIndeed, the C\\'ea lemma proves that $\\norm{u-U_\\circ}{H^{1\/2}(\\Gamma)}\\lesssim\\norm{(1-J_\\circ)u}{H^{1\/2}(\\Gamma)}$.\nNote that $u\\in H^1(\\Gamma)$ due of the mapping properties of $\\mathfrak{W}$ and $\\mathfrak{K}'$ and the assumption that $\\phi\\in L^2(\\Gamma)$.\nTherefore, the localization argument \\cite[Lemma~2.3]{faermann2d} in combination with the Sobolev-seminorm estimate \\cite[Lemma~4.5]{resigabem} gives that \n\\begin{align*}\n\\norm{(1-J_\\circ)u}{H^{1\/2}(\\Gamma)}\\lesssim \\norm{h_\\circ^{1\/2}\\partial_\\Gamma (1-J_\\circ)u}{L^2(\\Gamma)}+\\norm{h_\\circ^{-1\/2} (1-J_\\circ) u}{L^2(\\Gamma)}.\n\\end{align*}\nProposition~\\ref{lem:Scott properties} implies that \n\\begin{align*}\n\\norm{(1-J_\\circ)u}{H^{1\/2}(\\Gamma)}\\lesssim \\norm{h_\\circ^{1\/2} }{L^\\infty(\\Gamma)} \\norm{u}{H^1(\\Gamma)} \\to 0\\quad \\text{as}\\quad\\norm{h_\\circ^{1\/2} }{L^\\infty(\\Gamma)}\\to 0.\n\\end{align*}\n\n\\noindent\n\\textbf{Step 2:} \nFor $\\varepsilon>0$, let ${\\mathcal K}_\\circ$ be as in Step~1. \nThe triangle inequality and discrete reliability~(E3) yield that \n\\begin{align*}\n\\norm{u-U_\\bullet}{H^{1\/2}(\\Gamma)}\\le\\norm{u-U_\\circ}{H^{1\/2}(\\Gamma)} + \\norm{U_\\circ-U_\\bullet}{H^{1\/2}(\\Gamma)}\\le \\varepsilon + \\widetilde\\eta_\\bullet\\lesssim\\varepsilon+\\eta_\\bullet.\n\\end{align*}\nFor $\\varepsilon\\to0$, we conclude reliability \\eqref{eq:reliable}.\n\n\n\\subsection{Efficiency in (\\ref{eq:reliable})}\n\\label{sec:efficient hyper}\nClearly, it suffices to bound the residual part ${\\rm res}_\\bullet$ of the estimator $\\eta_\\bullet$ by $(\\norm{h_\\bullet^{1\/2} \\partial_\\Gamma( u - U_\\bullet )}{L^2(\\Gamma)}^2\n + \\norm{h_\\bullet^{1\/2} ( \\phi - \\phi_\\bullet )}{L^2(\\Gamma)}^2)^{1\/2}$.\n To do so, we use the triangle inequality \n\\begin{align}\\label{eq:aux efficiency}\n{\\rm res}_\\bullet\n\\le\n\\norm{h_\\bullet^{1\/2}(1\/2-\\mathfrak{K}')(\\phi-\\phi_\\bullet)}{L^2(\\Gamma)}\n+\n\\norm{h_\\bullet^{1\/2}(1\/2-\\mathfrak{K}')\\phi-\\mathfrak{W}U_\\bullet}{L^2(\\Gamma)}\n\\end{align}\nand bound each of the two terms separately. \nTo control the first one, we apply the inverse inequality \\eqref{eq:inverse Kpr} and the approximation property \\cite[Theorem~4.1]{cp06}\n\\begin{align*}\n\\norm{h_\\bullet^{1\/2}(1\/2-\\mathfrak{K}')(\\phi-\\phi_\\bullet)}{L^2(\\Gamma)}\n&\\stackrel{\\eqref{eq:inverse Kpr}}\\lesssim \n\\norm{\\phi-\\phi_\\bullet}{H^{-1\/2}(\\Gamma)} + \\norm{h_\\bullet^{1\/2}(\\phi-\\phi_\\bullet)}{L^2(\\Gamma)}\n\\\\\n&\\stackrel{\\text{\\cite{cp06}}}\\lesssim\n \\norm{h_\\bullet^{1\/2}(\\phi-\\phi_\\bullet)}{L^2(\\Gamma)}.\n\\end{align*}\nFor the second term in \\eqref{eq:aux efficiency}, we use the inverse inequality~\\eqref{eq:inverse W}\n\\begin{align*}\n\\norm{h_\\bullet^{1\/2}(1\/2-\\mathfrak{K}')\\phi-\\mathfrak{W}U_\\bullet}{L^2(\\Gamma)}\n&\\,\\,\\,=\\,\\,\\,\n\\norm{h_\\bullet^{1\/2}\\mathfrak{W}(u-U_\\bullet)}{L^2(\\Gamma)}\n\\\\\n&\\stackrel{\\eqref{eq:inverse W}}\\lesssim\n\\norm{u-U_\\bullet}{H^{1\/2}(\\Gamma)}\n+\n\\norm{h_\\bullet^{1\/2} \\partial_\\Gamma(u-U_\\bullet)}{L^2(\\Gamma)}. \n\\end{align*}\nAltogether, it only remains to estimate the term $\\norm{u-U_\\bullet}{H^{1\/2}(\\Gamma)}$. \nTo this end, we denote the Galerkin projection onto ${\\mathcal X}_\\bullet$ by $G_h:H^{1\/2}(\\Gamma)\\to{\\mathcal X}_\\bullet$ and note that $(1-G_\\bullet)=(1-G_\\bullet)(1-J_\\bullet)(1-G_\\bullet)$.\nThen, stability of $G_\\bullet$ and the approximation property~\\eqref{eq:local approx2} prove that\n\\begin{align*}\n\\norm{u-U_\\bullet}{H^{1\/2}(\\Gamma)}&\\,\\,\\,= \\,\\,\\,\n\\norm{(1-G_\\bullet)(1-J_\\bullet)(1-G_\\bullet)u}{H^{1\/2}(\\Gamma)}\n\\lesssim \\norm{(1-J_\\bullet)(1-G_\\bullet)u}{H^{1\/2}(\\Gamma)}\n\\\\\n&\\stackrel{\\eqref{eq:local approx2}}\\lesssim \\norm{h_\\bullet^{1\/2} (1-G_\\bullet) u}{L^2(\\Gamma)}=\n\\norm{h_\\bullet^{1\/2} \\partial_\\Gamma(u-U_\\bullet)}{L^2(\\Gamma)}. \n\\end{align*}\n\n\\subsection{General quasi-orthogonality (E4)}\n\\label{subsec:hypsing orthogonality}\nFor sufficiently small $\\vartheta>0$, we prove general quasi-orthogonality in three steps.\nFor ${\\mathcal K}_\\bullet,{\\mathcal K}_\\star\\in\\mathbb{K}$, we define ${\\mathcal K}_{\\bullet\\cap\\star}\\in\\mathbb{K}$ via ${\\mathcal N}_{\\bullet\\cap\\star}:={\\mathcal N}_\\bullet\\cap{\\mathcal N}_\\star$ and $\\#_{\\bullet\\cap\\star} z:=\\min\\{\\#_{\\bullet}z,\\#_\\star z\\}$ for all $z\\in{\\mathcal N}_\\bullet\\cap{\\mathcal N}_\\star$.\n\n\\noindent\n\\textbf{Step 1:}\nFirst, we prove some kind of discrete reliability of $\\mu$:\nThere exists a constant $C_{\\rm drel}^-\\ge1$ \nsuch that %\n\\begin{align}\\label{eq:drel}\n\\norm{U_{k}-U_{k\\cap (k+1)}}{H^{1\/2}(\\Gamma)}^{2}\\le C_{\\rm drel}^-\\, \\mu_k(\\mathcal M_k^-)^{2} \\quad \\text{for all }k\\in{\\mathbb N}_0.\n\\end{align}\nTo see this, we note that $U_{k\\cap (k+1)}\\in{\\mathcal X}_k\\cap{\\mathcal X}_{k+1}$ is also the Galerkin projection of $U_k$.\nHence, the C\\'ea lemma and the inverse estimate \\eqref{eq:discrete invest2} yield that\n\\begin{align*}\n\\norm{U_{k}-U_{k\\cap (k+1)}}{H^{1\/2}(\\Gamma)}^2&\\stackrel{\\text{C\\'ea}}\\lesssim \\norm{(1-J_{k\\cap (k+1)})U_{k}}{H^{1\/2}(\\Gamma)}^2\\stackrel{\\eqref{eq:discrete invest2}}\\lesssim \\norm{h_{k\\cap (k+1)}^{-1\/2}(1-J_{k\\cap (k+1)})U_{k}}{L^2(\\Gamma)}^2.\n\\end{align*}\nNote that $h_{k}=h_{k\\cap (k+1)}$ and $\\pi_{k}(\\cdot)=\\pi_{k\\cap (k+1)}(\\cdot)$.\nFurther, Lemma~\\ref{lem:properties for B-splines} \\eqref{item:spline basis} shows for all $Q\\in{\\mathcal{Q}}_k$ that $U_k|_{\\pi_k^p(Q)}\\in{\\mathcal X}_{k\\cap (k+1)}|_{\\pi_k^p(Q)}$ if $\\pi_k^p(Q)\\cap\\mathcal M_k^-=\\emptyset$.\nThus, the local projection property \\eqref{eq:local projection} yields that\n\\begin{align*}\n\\norm{h_{k\\cap (k+1)}^{-1\/2}(1-J_{k\\cap (k+1)})U_{k}}{L^2(\\Gamma)}^2=\\norm{h_{k}^{-1\/2}(1-J_{k\\cap (k+1)})U_{k}}{L^2(\\pi_{k}^{p+1}(\\mathcal M_{k}^-))}^2.\n\\end{align*}\nNote that ${\\mathcal X}_{k\\ominus1}\\subseteq{\\mathcal X}_{k\\cap (k+1)}$.\nTogether with the projection property \\eqref{eq:local projection} and the local $L^2$-stability \\eqref{eq:local L2}, the triangle inequality implies that\n\\begin{align*}\n&\\norm{h_{k}^{-1\/2}(1-J_{k\\cap (k+1)})U_{k}}{L^2(\\pi_{k}^{p+1}(\\mathcal M_{k}^-))}\n\\le \\norm{h_{k}^{-1\/2}(1-J_{k\\cap (k+1)}J_{k-\\ominus 1})U_{k}}{L^2(\\pi_{k}^{p+1}(\\mathcal M_{k}^-))} \n\\\\&\\hspace{7cm}+\\norm{h_{k}^{-1\/2}J_{k\\cap(k+1)}(1-J_{\\ominus1})U_{k}}{L^2(\\pi_{k}^{p+1}(\\mathcal M_{k}^-))}^2\n\\\\\n&\\stackrel{\\eqref{eq:local projection}+\\eqref{eq:local L2}}\\lesssim \\norm{h_{k}^{-1\/2}(1-J_{k\\ominus 1})U_{k}}{L^2(\\pi_{k}^{2p+1}(\\mathcal M_{k}^-))}\\le \\mu_k(\\mathcal M_{k}^-).\n\\end{align*}\nThe constant $\\const{drel}^-$ in \\eqref{eq:drel} depends only on the parametrization $\\gamma$, the polynomial order $p$, and the initial mesh $\\widehat{\\mathcal{Q}}_0$.\n\n\n\\noindent\\textbf{Step 2:}\nNext, we prove \nthe existence of some constant $C_{\\rm mon}'\\ge 1$ such that\n\\begin{align}\\label{eq:qquasi monotonicity}\n\\widetilde\\eta_{k+1}^{\\,2}\\le C_{\\rm mon}'\\, \\widetilde\\eta_k^{\\,2}\\quad\\text{for all }k\\in{\\mathbb N}_0.\n\\end{align}\nBy stability (E1) and reduction (E2), we have that\n\\begin{align}\\label{eq:monoton help}\n\\widetilde\\eta_{k+1}^{\\,2}\\lesssim \\widetilde\\eta_k^{\\,2}+\\norm{U_{k+1}-U_k}{H^{1\/2}(\\Gamma)}^2+\\norm{\\phi_{k+1}-\\phi_k}{H^{-1\/2}(\\Gamma)}^2.\n\\end{align}\nTo estimate $\\norm{U_{k+1}-U_k}{H^{1\/2}(\\Gamma)}^2$ of \\eqref{eq:monoton help}, we use ellipticity, Galerkin orthogonality, and Young's inequalit\n\\begin{align}\n&\\norm{U_{k+1}-U_k}{H^{1\/2}(\\Gamma)}^2\\simeq\\norm{U_{k+1}-U_k}{\\mathfrak{W}}^2\\lesssim \n\\norm{U_{k+1}-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2+\n\\norm{U_k-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2\\notag\\\\\n\\label{eq:k+1-k}\n&\\quad=\n\\big(\\norm{u-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2\n-\\norm{u-U_{k+1}}{\\mathfrak{W}}^2\\big)\n+\\norm{U_k-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2\\\\\n&\\quad\\le 2\\norm{u-U_k}{\\mathfrak{W}}^2+3 \\norm{U_k-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2.\n\\stackrel{\\eqref{eq:drel}}\n\\lesssim \\norm{u-U_k}{\\mathfrak{W}}^2+\\const{drel}^-\\,\\mu_k(\\mathcal M_k^-)^2\n\\stackrel{\\eqref{eq:reliable}+\\rm(v)}\\lesssim \\eta_k^2\n\\stackrel{\\eqref{eq:abstract equivalence}}\\simeq\\widetilde\\eta_k^{\\,2}.\\notag\n\\end{align}\nTo estimate $\\norm{\\phi_{k+1}-\\phi_k}{H^{-1\/2}(\\Gamma)}^2$ of \\eqref{eq:monoton help}, we note that (although ${\\mathcal X}_k$ and ${\\mathcal X}_{k+1}$ are not necessarily nested) the set of (transformed) ${\\mathcal{Q}}_k$-piecewise polynomials of degree $p$ is a subset of the set of (transformed) ${\\mathcal{Q}}_{k+1}$-piecewise polynomials of degree $p$.\nHence, \\eqref{eq:proj proj} gives that\n\\begin{align}\\label{eq:E4 E3 connect}\n\\begin{split}\n\\norm{\\phi_{k+1}-\\phi_k}{H^{-1\/2}(\\Gamma)} \n\\stackrel{\\eqref{eq:proj proj}}\\lesssim\n \\norm{h_k^{1\/2}(\\phi- \\phi_k)}{L^2(\\bigcup ({\\mathcal{Q}}_k\\setminus{\\mathcal{Q}}_{k+1}))}\n\\le {\\rm osc}_k \\stackrel{\\eqref{eq:tilde h equivalent}}\\simeq\\widetilde{\\rm osc}_k.\n\\end{split}\n\\end{align}\nThe constant $C_{\\rm mon}'$ depends only on the parametrization $\\gamma$, the polynomial order $p$, the initial mesh $\\widehat{\\mathcal{Q}}_0$, and an arbitrary but fixed upper bound for the parameter $\\vartheta$.\n\\\\\n\\textbf{Step 3:} We finally come to (E4) itself.\nWith \\eqref{eq:k+1-k}, Galerkin orthogonality gives that\n\\begin{align}\n\\label{eq:second Galerkin}\n&\\norm{U_{k+1}-U_k}{H^{1\/2}(\\Gamma)}^2\n\\stackrel{\\eqref{eq:k+1-k} }\\lesssim\n\\big(\\norm{u-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2\n-\\norm{u-U_{k+1}}{\\mathfrak{W}}^2\\big)\n+\\norm{U_k-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2\\\\\n&= \\norm{u-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2-\\norm{u-U_{(k+1) \\cap (k+2)}}{\\mathfrak{W}}^2\n+\\norm{U_{k+1}-U_{(k+1) \\cap (k+2)}}{\\mathfrak{W}}^2+\\norm{U_k-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2.\n\\notag\n\\end{align}\nWe abbreviate the hidden (generic) constant by $C>0$.\nWith Step 1 and 2 in combination with\nAlgorithm~\\ref{the algorithm}~(v),\nthe third plus the fourth term can be estimated by\n\\begin{align}\n\\begin{split}\n&\n\\norm{U_{k+1}-U_{(k+1) \\cap (k+2)}}{\\mathfrak{W}}^2+\\norm{U_k-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2\n\\stackrel{\\eqref{eq:drel}}\n\\le \nC_{\\rm drel}^-\\big(\\mu_{k+1}(\\mathcal M_{k+1}^-)+\\mu_k(\\mathcal M_k^-)\\big)\n\\\\\n&\\qquad\\stackrel{{\\rm (v)}}\\le \nC_{\\rm drel}^-\\vartheta(\\eta_{k+1}^2+\\eta_k^2)\n\\stackrel{\\eqref{eq:qquasi monotonicity}}\\le \n C_{\\rm drel}^-C_{\\rm eq}\\vartheta(\\widetilde\\eta_{k+1}^{\\,2}+\\widetilde\\eta_k^{\\,2})\n \\le\n C_{\\rm drel}^- C_{\\rm eq}(C_{\\rm mon}'+1)\\vartheta\\widetilde\\eta_k^{\\,2}.\n \\end{split}\n\\end{align}\nSuppose that\n$\\vartheta>0$ is sufficiently small such that \n\\begin{align}\n\\label{eq:E4 vartheta}\n\\varepsilon_{\\rm qo}:=\nC C_{\\rm drel}^- C_{\\rm eq}(C_{\\rm mon}'+1)\\vartheta\n <\n \\sup_{\\widetilde\\delta>0}\\frac{1-(1+\\widetilde\\delta)(1-(1-\\rho_{\\rm red})\\theta)}{C_{\\rm red}+(2+\\widetilde\\delta^{\\,-1})C_{\\rm stab}^2}.\n\\end{align} \nCombining \\eqref{eq:second Galerkin}--\\eqref{eq:E4 vartheta}, we obtain that\n\\begin{align*}\n\\norm{U_{k+1}-U_k}{H^{1\/2}(\\Gamma)}^2-\\varepsilon_{\\mathrm{qo}}\\widetilde\\eta_k^{\\,2}\n\\lesssim \n\\norm{u-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2-\\norm{u-U_{(k+1) \\cap (k+2)}}{\\mathfrak{W}}^2.\n\\end{align*}\n \nTogether with Step 1, \nAlgorithm~\\ref{the algorithm}~(v) and reliability \\eqref{eq:reliable}, we derive that \n\\begin{align*}\n&\\sum_{k=\\ell}^{\\ell+N}\\Big(\\norm{U_{k+1}-U_k}{H^{1\/2}(\\Gamma)}^2-\\varepsilon_{\\mathrm{qo}}\\widetilde\\eta_k^{\\,2}\\Big)\n\\lesssim \n\\sum_{k=\\ell}^{\\ell+N} \\Big(\\norm{u-U_{k\\cap (k+1)}}{\\mathfrak{W}}^2-\\norm{u-U_{(k+1) \\cap (k+2)}}{\\mathfrak{W}}^2\\Big)\n\\\\\n&\\,\\leq\\, \\norm{u-U_{\\ell\\cap (\\ell+1)}}{\\mathfrak{W}}^2\\lesssim \\norm{u-U_\\ell}{\\mathfrak{W}}^2+\\norm{U_\\ell-U_{\\ell\\cap (\\ell+1)}}{\\mathfrak{W}}^2\n\\stackrel{\\eqref{eq:drel}}\\lesssim \\norm{u-U_\\ell}{\\mathfrak{W}}^2+\\mu_k(\\mathcal M_k^-)^{2} \n\\\\\n&\\stackrel{{\\rm (v)}}\\lesssim \n\\norm{u-U_\\ell}{\\mathfrak{W}}^2+\\vartheta\\widetilde\\eta_\\ell^{\\,2}\n\\stackrel{\\eqref{eq:abstract equivalence}}\\lesssim\n\\widetilde\\eta_\\ell^{\\,2}\n\\stackrel{\\eqref{eq:reliable}}\\simeq\n\\eta_\\ell^2.\n\\end{align*} \n\nIt remains to estimate the sum $\\sum_{k=\\ell}^{\\ell+N}\\norm{\\phi_{k+1}-\\phi_k}{H^{-1\/2}(\\Gamma)}^2$. \nTo this end, we note that $h_{k+1}\\le q h_k$ on $\\bigcup({\\mathcal{Q}}_k\\setminus{\\mathcal{Q}}_{k+1})$ for some constant $00$ such that\n\\begin{align}\\label{eq:estimator reduction}\n\\widetilde\\eta_{\\ell+1}^{\\,2}\\le \\ro{est}\\,\\widetilde\\eta_\\ell^{\\,2}+\\const{est} \\varrho_{\\ell,\\ell+1}^2\\quad\\text{for all }\\ell\\in{\\mathbb N}_0,\n\\end{align}\nwhere $\\ro{est}=(1+\\delta)(1-(1-\\ro{red})\\widetilde\\theta)$ with $\\widetilde\\theta:=\\const{eq}^{-2}\\theta$ and $\\const{est}=\\const{red}+(1+\\delta^{-1})\\const{stab}^2$ for all sufficiently small $\\delta>0$ with $\\ro{est}<1$. \nThe critical observation is that Algorithm~\\ref{the algorithm} implies that \n$\\mathcal M_\\ell\\subseteq {\\mathcal N}^{\\rm ref}_{\\ell,\\ell+1}$, where $\\mathcal M_\\ell$ satisfies the D\\\"orfler marking $\\theta\\eta_\\ell^2\\le\\eta_\\ell(\\mathcal M_\\ell)^2$ and thus $\\widetilde\\theta\\widetilde\\eta_\\ell^{\\,2}\\le\\widetilde\\eta_\\ell(\\mathcal M_\\ell)^{\\,2}$ due to the equivalence \\eqref{eq:abstract equivalence}.\nWith this, the proof follows along the lines of \\cite[Section~4.3]{axioms}.\nWe split the estimator, apply the Young inequality in combination with stability (E1) and reduction (E2) to see that, for all $\\delta>0$, \n\\begin{align*}\n\\widetilde\\eta_{\\ell+1}^{\\,2}&=\\widetilde\\eta_{\\ell+1}({\\mathcal N}^{\\rm id}_{\\ell,\\ell+1})^{\\,2}+\\widetilde\\eta_{\\ell+1}({\\mathcal N}^{\\rm ref}_{\\ell+1,\\ell})^{\\,2}\\\\\n&\\le (1+\\delta)\\widetilde\\eta_{\\ell}({\\mathcal N}^{\\rm id}_{\\ell,\\ell+1})^{\\,2}\n+\\ro{red}\\widetilde\\eta_{\\ell}({\\mathcal N}^{\\rm ref}_{\\ell,\\ell+1})^{\\,2}\n+\\const{est}\\,\\varrho_{\\ell,\\ell+1}^2\\\\\n&\\le (1+\\delta)\\big(\\widetilde\\eta_\\ell^{\\,2}-(1-\\ro{red})\\big) \\,\\widetilde\\eta_\\ell({\\mathcal N}^{\\rm id}_{\\ell,\\ell+1})^{\\,2}\n+\\const{est}\\varrho_{\\ell,\\ell+1}^2\\\\\n&\\le (1+\\delta)(1-(1-\\ro{red})\\widetilde\\theta)\\,\\widetilde\\eta_\\ell^{\\,2}+\\const{est}\\,\\varrho_{\\ell,\\ell+1}^2, \n\\end{align*}\nwhich concludes estimator reduction \\eqref{eq:estimator reduction}. \nAccording to \\cite[Proposition~4.10]{axioms}, this together with general quasi-orthogonality~(E4) yields linear convergence of $\\widetilde\\eta$ and thus also of $\\eta$ due to the equivalence~\\eqref{eq:abstract equivalence}.\n\n\n\n\\subsection{Optimal convergence (\\ref{eq:optimal})}\n\\label{sec:optimal convergence}\nWe start with the following proposition, which states that D\\\"orfler marking is not only sufficient for linear convergence, but in some sense even necessary.\nFor standard element-based adaptive algorithms, it is proved, e.g., in \\cite[Proposition~4.12]{axioms}.\nWe note that the proof follows essentially along the same lines and is only given for the sake of completeness.\n\n \\begin{proposition}\\label{prop:Doerfler optimal}\nSuppose stability {\\rm (E1)} and discrete reliability {\\rm (E3)}.\nLet ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ and ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$.\nThen, for all $0<\\widetilde\\theta<\\widetilde\\theta_{\\rm opt}:=(1+\\const{stab}^2\\const{drel}^2)^{-1}$, there exists some $0<\\ro{\\widetilde\\theta}<1$ such that \n\\begin{align}\\label{eq:Doerfler optimal}\n\\widetilde\\eta_\\circ^{\\,2}\\le \\ro{\\widetilde\\theta}\\,\\widetilde\\eta_\\bullet^{\\,2}\n\\quad \\Longrightarrow \\quad \\widetilde\\theta\\,\\widetilde\\eta_\\bullet^{\\,2}\\le \\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ})^{\\,2}.\n\\end{align}\nThe constant $\\ro{\\widetilde\\theta}$ depends only on $\\const{stab}, \\const{drel}$, $\\widetilde\\theta$.\n \\end{proposition}\n \\begin{proof}\n Throughout the proof, we work with a free variable $\\ro{\\widetilde\\theta}>0$, which will be fixed at the end.\nFor all $\\delta>0$, the Young inequality together with stability (E1) shows that\n \\begin{align*}\n \\widetilde\\eta_\\bullet^{\\,2}=\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ})^2 +\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})^2\\le \\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ})^2+(1+\\delta^{-1}) \\widetilde\\eta_\\circ({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})^2+(1+\\delta)C_{\\rm stab}^2\\,\\varrho_{\\bullet,\\circ}^2.\n \\end{align*}\nWith $\\RR_{\\bullet,\\circ}\\supseteq{\\mathcal N}^{\\rm ref}_{\\bullet,\\circ}$, we get for the first term on the right-hand side that \n$\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ})^2\\le \\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ})^2$.\nThe assumption \\eqref{eq:Doerfler optimal} proves that $\\widetilde\\eta_\\circ({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})^2\\le \\widetilde\\eta_\\circ^{\\,2}\\le \\ro{\\widetilde\\theta}\\,\\widetilde\\eta_\\bullet^{\\,2}$.\nTogether with discrete reliability (E3), we obtain that \n\\begin{align*}\n\\widetilde\\eta_\\bullet^{\\,2}\\le \\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ})^2 +(1+\\delta^{-1}) \\ro{\\widetilde\\theta}\\,\\widetilde\\eta_\\bullet^{\\,2}+(1+\\delta)C_{\\rm stab}^2\\const{drel}^2\\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ})^2.\n\\end{align*}\nPut differently, we end up with\n\\begin{align*}\n\\frac{1-(1+\\delta^{-1})\\ro{\\widetilde\\theta}}{1+(1+\\delta)\\const{stab}^2\\const{drel}^2}\\widetilde\\eta_\\bullet^{\\,2}\\le \\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ})^2.\n\\end{align*}\nFinally, we choose $\\delta>0$ and then $0<\\ro{\\widetilde\\theta}<1$ such that \n\\begin{align*}\n\\widetilde\\theta\\le \\frac{1-(1+\\delta^{-1})\\ro{\\widetilde\\theta}}{1+(1+\\delta)\\const{stab}^2\\const{stab}^2\\const{drel}^2}<\\frac{1}{1+\\const{drel}^2}=\\widetilde\\theta_{\\rm opt}.\n\\end{align*}\nThis concludes the proof.\n\\end{proof}\n\n\n\n\nIn the following lemma, we show that the estimator is monotone up to some multiplicative constant.\nAgain, the proof follows along the lines of the version from \\cite[Lemma~3.5]{axioms}. \n \\begin{lemma}\\label{lem:quasi-monotonicity}\nSuppose {\\rm (E1)--(E3)}, where the restriction $\\ro{red}<1$ is not necessary.\nThen, there exists a constant $\\const{mon}\\ge1$ such that there holds quasi-monotonicity in the sense that \n\\begin{align}\\label{eq:quasi-monotonicity}\n\\widetilde\\eta_\\circ^{\\,2}\\le \\const{mon} \\widetilde\\eta_\\bullet^{\\,2} \\quad\\text{for all } {\\mathcal K}_\\bullet\\in\\mathbb{K}, {\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet).\n\\end{align}\nThe constant $\\const{mon}$ depends only on $\\const{stab},\\const{red}$, $\\ro{red}$, and $\\const{drel}$.\n \\end{lemma}\n \\begin{proof}\n We split the estimator and apply Young's inequality in combination with (E1)--(E2).\nFor all $\\delta>0$, we see that\n \\begin{align*}\n \\widetilde\\eta_\\circ^{\\,2}=\\widetilde\\eta_\\circ({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})^2+\\widetilde\\eta_\\circ({\\mathcal N}^{\\rm ref}_{\\circ,\\bullet})^2&\\le (1+\\delta)\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm id}_{\\bullet,\\circ})^2+\\ro{red}\\widetilde\\eta_\\bullet({\\mathcal N}^{\\rm ref}_{\\bullet,\\circ})^2+(\\const{stab}^2+\\const{red}(1+\\delta^{-1}))\\varrho_{\\bullet,\\circ}^2\\\\\n &\\le \\max\\{1+\\delta,\\ro{red}\\} \\widetilde\\eta_\\bullet^{\\,2}+(\\const{stab}^2+\\const{red}(1+\\delta^{-1}))\\varrho_{\\bullet,\\circ}^2.\n \\end{align*}\n The application of (E3) yields that \n \\begin{align*}\n \\widetilde\\eta_\\circ^{\\,2}&\\le \\max\\{1+\\delta,\\ro{red}\\}\\widetilde\\eta_\\bullet^{\\,2}+(\\const{stab}^2+\\const{red}(1+\\delta^{-1}))\\const{drel}^2\\widetilde\\eta_\\bullet(\\RR_{\\bullet,\\circ})^2\\\\\n &\\le\\big(\\max\\{1+\\delta,\\ro{red}\\}+(\\const{stab}^2+\\const{red}(1+\\delta^{-1}))\\const{drel}^2\\big)\\widetilde\\eta_\\bullet^{\\,2}.\n \\end{align*}\nThis concludes the proof.\n \\end{proof}\n\n The next lemma provides the key ingredient for the proof of optimal convergence rates.\n Again, the proof follows along the lines of \\cite[Lemma~4.14]{axioms}. \n\n \\begin{lemma}\\label{lem:optimality}\nSuppose the overlay property {\\rm (R3)} and quasi-monotonicity \\eqref{eq:quasi-monotonicity}. \nLet $\\ell\\in{\\mathbb N}_0$ such that $\\widetilde\\eta_\\ell>0$ and let $0<\\ro{}<1$. \nLet $s>0$ with $\\norm{u}{\\widetilde{\\mathbb{A}}_s}:=\\sup_{N\\in{\\mathbb N}_0} \\big( (N+1)^s\\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}(N)} \\widetilde\\eta_\\bullet\\big)<\\infty$.\n\tThen, there exists a refinement ${\\mathcal K}_{\\circ}\\in{\\tt refine}({\\mathcal K}_\\ell)$ with\n\t%\n\t\\begin{subequations}\\label{eq:lemoptresult}\n\t\t\\begin{align}\n\t\t\\label{eq:lemoptresult1}\n\t\t\\widetilde\\eta_{\\circ}^{\\,2} &\\leq \\ro{} \\,\\widetilde\\eta_\\ell^{\\,2},\n\t\t\\\\\n\t\t\t\t\\label{eq:lemoptresult2}\n\t\t\\#_\\circ{\\mathcal N}_\\circ-\\#_\\ell{\\mathcal N}_\\ell &<\\const{mon}^{1\/(2s)}\\norm{u}{\\widetilde{\\mathbb{A}}_s}^{1\/s}\\,\\ro{}^{-1\/(2s)}\\widetilde\\eta_\\ell^{\\,-1\/s}.\n\t\t\\end{align}\n\t\t\\end{subequations} \n\\end{lemma}\n\n\\begin{proof}\nWe prove the assertion in two steps.\n\n\\noindent\\textbf{Step 1:} \nWe show a modified \\eqref{eq:lemoptresult} for some ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ instead of a refinement ${\\mathcal K}_\\circ\\in{\\tt refine}({\\mathcal K}_\\bullet)$, i.e., we prove\nthat\n\\begin{subequations}\n\t\t\\begin{align}\t\t\n\t\t\\label{eq:lemoptresult3}\n\t\t\\widetilde\\eta_{\\bullet}^{\\,2} &\\leq {(\\ro{}}\/\\const{mon}) \\,\\widetilde\\eta_\\ell^{\\,2},\n\t\t\\\\\n\t\t\\label{eq:lemoptresult4}\n\t\t\\#_\\bullet{\\mathcal N}_{\\bullet}-\\#_0{\\mathcal N}_0 &<\\norm{u}{\\widetilde{\\mathbb{A}}_s}^{1\/s}\\,(\\ro{}\/\\const{mon})^{-1\/(2s)}\\widetilde\\eta_\\ell^{\\,-1\/s}.\n\t\t\\end{align}\n\t\t\t\t\\end{subequations}\nLet $N\\in{\\mathbb N}_0$ be minimal such that $\\norm{u}{\\widetilde{\\mathbb{A}}_s}(N+1)^{-s} \\le {(\\ro{}\/\\const{mon})}^{\\,1\/2}\\widetilde\\eta_\\ell$.\n\tNote that $N>0$ by the fact that $\\widetilde\\eta_\\ell\\leq\\const{mon}^{1\/2}\\widetilde\\eta_0\\le\\const{mon}^{1\/2}\\norm{u}{\\widetilde{\\mathbb{A}}_s}$ and $00$.\nClearly, with the equivalence \\eqref{eq:abstract equivalence}, this immediately gives \\eqref{eq:optimal}.\nWithout loss of generality, we assume that $\\norm{u}{\\widetilde{\\mathbb{A}}_s}<\\infty$.\nIf $\\widetilde\\eta_{\\ell_0}=0$ for some $\\ell_0\\in{\\mathbb N}_0$, then, \nAlgorithm~\\ref{the algorithm} implies that $\\widetilde\\eta_\\ell=0$ for all $\\ell\\geq \\ell_0$.\nMoreover, $(\\#_0{\\mathcal N}_0-\\#_0{\\mathcal N}_0+1)^s\\widetilde\\eta_0\\le \\norm{u}{\\widetilde{\\mathbb{A}}_s}$ is trivially satisfied.\nThus, it is sufficient to consider $0<\\ell< \\ell_0$ resp.\\ $0<\\ell$ if no such $\\ell_0$ exists.\nNow, let $k< \\ell$ and define $\\widetilde\\theta:=\\const{eq}^{2}\\theta$.\nAccording to Lemma~\\ref{lem:quasi-monotonicity}, we may apply Lemma~\\ref{lem:optimality} for ${\\mathcal K}_k$, where we choose $\\ro{\\widetilde\\theta}$ as in Proposition~\\ref{prop:Doerfler optimal}.\nIn particular, \\eqref{eq:Doerfler optimal} in combination with \\eqref{eq:lemoptresult1} shows that $\\RR_{k,\\circ}$ satisfies the D\\\"orfler marking $\\widetilde\\theta\\widetilde\\eta_k^{\\,2}\\le\\widetilde\\eta_k(\\RR_{k,\\circ})^2$ and hence $\\theta\\eta_k^2\\le\\eta_k(\\RR_{k,\\circ})^2$.\nSince, $\\mathcal M_k$ is an essentially minimal set satisfying D\\\"orfler marking (see Remark~\\ref{rem:feature} {\\rm (a)}) we get that\n$|\\mathcal M_k|\\lesssim|\\RR_{k,\\circ}|$.\nSince the maximal multiplicity is bounded, we see that \n\\begin{align*}\n\\#_k\\mathcal M_k\\lesssim\\#_k\\RR_{k,\\circ}\\stackrel{\\rm (E3)}\\lesssim\n\t \\#_\\circ{\\mathcal N}_\\circ-\\#_k{\\mathcal N}_k\n\\stackrel{\\eqref{eq:lemoptresult2}}\\lesssim\n\\norm{u}{\\widetilde{\\mathbb{A}}_s}^{1\/s}\\,\n\t \\widetilde\\eta_k^{\\,-1\/s}.\n\t\\end{align*}\nFor $\\ell>0$, the closure estimate (R2) proves that\n\t\\begin{align*}\n\t\\begin{split}\n\t\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0+1\\le 2(\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0)\n\t\\leq 2\\const{clos} \\sum_{k=0}^{\\ell-1}\\#_k\\mathcal M_k\n\t\\leq \n\n\t\\norm{u}{\\widetilde{\\mathbb{A}}_s}^{1\/s}\n\n\t\\sum_{k=0}^{\\ell-1}\\widetilde\\eta_k^{\\,-1\/s}.\n\t\\end{split}\n\t\\end{align*}\n\tFinally, linear convergence of $\\eta_k$ \\eqref{eq:R-linear} and thus of $\\widetilde \\eta_k$ and elementary analysis show that the term $\\sum_{k=0}^{\\ell-1}\\widetilde\\eta_k^{\\,-1\/s}$ can be bounded from above by $C\\widetilde\\eta_\\ell^{\\,-1\/s}$ where $C>0$ depends only on $\\ro{lin}, \\const{lin}$, and $s$.\nTherefore,\twe end up with \n\t\\begin{align*}\n\t(\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0+1)^{s}\\widetilde\\eta_\\ell\\lesssim \n\\norm{u}{\\widetilde{\\mathbb{A}}_s}\\quad\\text{for all }\\ell>0.\n\t\\end{align*}\nFor $\\ell=0$, the latter estimate is trivially satisfied. This concludes the proof.\n\n\\noindent \\textbf{Step 2:}\nTo see the lower bound in \\eqref{eq:optimal}, let $N\\in{\\mathbb N}_0$ and choose the maximal $\\ell\\in{\\mathbb N}_0$ such that $\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0\\le N$.\nDue to the maximality of $\\ell$ and the son estimate (R1), we have that \n$N+1\\le\\#_{\\ell+1}{\\mathcal N}_{\\ell+1}-\\#_0{\\mathcal N}_\n\\le \\const{son}\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0\n\\simeq \\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0+1$,\nwhere the hidden constants depend only on $\\#_0{\\mathcal N}_0$.\nThis leads to\n\\begin{align*}\n\\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}(N)} (N+1)^s\\eta_\\bullet\\lesssim (\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0+1)^s\\eta_\\ell\n\\end{align*}\nand concludes the proof.\n\n\\end{proof}\n\n\n\\subsection{Approximability constants satisfy (\\ref{eq:classes})}\n\\label{sec:classes}\nThe second inequality in \\eqref{eq:classes} is trivially satisfied by definition of the approximability constants and the fact that $\\mathbb{K}^1\\cup\\mathbb{K}^p\\subseteq\\mathbb{K}$. \n\nFor the first inequality, we call Algorithm~\\ref{the algorithm} with parameters as in Remark~\\ref{rem:feature}~(c) such that only $h$-refinement takes place.\n\\eqref{eq:optimal} gives that\n$\\norm{u}{\\mathbb{A}_s}$ $\\simeq$ $\\sup_{\\ell\\in{\\mathbb N}_0}{(\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0+1)^{s}}{\\eta_\\ell}$. \nSince ${\\mathcal K}_\\ell\\in\\mathbb{K}^1$ for all $\\ell\\in{\\mathbb N}_0$, we can argue along the lines of Step~2 of the proof of~\\eqref{eq:optimal} to see that $\\norm{u}{\\mathbb{A}_s^1}\\lesssim\\sup_{\\ell\\in{\\mathbb N}_0}{(\\#_\\ell{\\mathcal N}_\\ell-\\#_0{\\mathcal N}_0+1)^{s}}{\\eta_\\ell}$.\n\nFor the third inequality in \\eqref{eq:classes}, we note the elementary equivalence for arbitrary fixed constants $C>0$\n\\begin{align*}\n\\norm{u}{\\mathbb{A}^p_s}=\\sup_{N\\in{\\mathbb N}_0}\\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}^p(N)} (N+1)^s \\eta_{\\bullet}\\simeq\\sup_{N\\in{\\mathbb N}_0}\\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}^p(CN)}(N+1)^s \\eta_{\\bullet}.\n\\end{align*}\nTo conclude the proof of \\eqref{eq:classes}, it thus remains to show that \n\\begin{align*}\n\\sup_{N\\in{\\mathbb N}_0}\\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}^p(CN)} \\eta_{\\bullet}\\lesssim\\sup_{N\\in{\\mathbb N}_0} \\inf_{{\\mathcal K}_\\bullet\\in\\mathbb{K}^1(N)} \\eta_{\\bullet}\n\\end{align*}\nfor some generic constant $C>0$.\nLet $N\\in{\\mathbb N}_0$. To verify the latter inequality, let ${\\mathcal K}_{\\bullet,1}\\in\\mathbb{K}^1(N)$.\nMoreover, let ${\\mathcal K}_{\\bullet,p}$ be the corresponding knots in $\\mathbb{K}^p$ with ${\\mathcal N}_{\\bullet,p}={\\mathcal N}_{\\bullet,1}$.\nRecall the initial knots ${\\mathcal K}_{0,p}$ with maximal multiplicity $p$ from \\eqref{eq:Ks}. \nIf ${\\mathcal K}_{\\bullet,1}\\neq{\\mathcal K}_0$ and thus $\\#_{\\bullet,1}{\\mathcal N}_{\\bullet,1}>\\#_0{\\mathcal N}_0$, there exists a constant $C>0$ only depending on $p$ and $|{\\mathcal N}_0|$, such that\n\\begin{align*}\n\\#_{\\bullet,p}{\\mathcal N}_{\\bullet,p}-\\#_{0,p}{\\mathcal N}_{0,p}\n= p(|{\\mathcal N}_{\\bullet,1}|-|{\\mathcal N}_0|)\n\\le p( \\#_{\\bullet,1}{\\mathcal N}_{\\bullet,1}-|{\\mathcal N}_0|)\n\\le C( \\#_{\\bullet,1}{\\mathcal N}_{\\bullet,1}-\\#_0{\\mathcal N}_0) \\le C N,\n\\end{align*}\nwhich yields that ${\\mathcal K}_{\\bullet,1}\\in\\mathbb{K}^1(N)$. \nTo conclude the proof, it is thus remains to show that $\\eta_{\\bullet,p}\\lesssim \\eta_{\\bullet,1}$. \nSince $\\phi_{\\bullet,p}=\\phi_{\\bullet,1}$, we have that ${\\rm osc}_{\\bullet,p}={\\rm osc}_{\\bullet,1}$. \nFor the residual term, we note that $h_{\\bullet,p}=h_{\\bullet,1}$ and $g_{\\bullet,p}=g_{\\bullet,1}$.\nThe triangle inequality gives that\n\\begin{align*}\n{\\rm res}_{\\bullet,p}=\\norm{h_{\\bullet,p}^{1\/2}(g_{\\bullet,p}-\\mathfrak{W}U_{\\bullet,p})}{L^2(\\Gamma)}\n&\\le\n\\norm{h_{\\bullet,1}^{1\/2}(g_{\\bullet,1}-\\mathfrak{W}U_{\\bullet,1})}{L^2(\\Gamma)}\n+\\norm{h_{\\bullet,p}^{1\/2}\\mathfrak{W}(U_{\\bullet,p}-U_{\\bullet,1})}{L^2(\\Gamma)}\n\\\\\n&={\\rm res}_{\\bullet,1}\n+\\norm{h_{\\bullet,p}^{1\/2}\\mathfrak{W}(U_{\\bullet,p}-U_{\\bullet,1})}{L^2(\\Gamma)}.\n\\end{align*}\nTo estimate the second summand, we note that ${\\mathcal K}_{\\bullet,p}\\in{\\tt refine}({\\mathcal K}_{\\bullet,1})$.\nTherefore, we can use the inverse inequalities \\eqref{eq:inverse W} and \\eqref{eq:discrete invest} and discrete reliability (E3) to see that\n\\begin{align*}\n\\norm{h_{\\bullet,p}^{1\/2}\\mathfrak{W}(U_{\\bullet,p}-U_{\\bullet,1})}{L^2(\\Gamma)}\n&\\stackrel{\\eqref{eq:inverse W}}\\lesssim\n\\norm{U_{\\bullet,p}-U_{\\bullet,1}}{H^{1\/2}(\\Gamma)} + \\norm{h_\\bullet^{1\/2}\\partial_\\Gamma (U_{\\bullet,p}-U_{\\bullet,1})}{L^2(\\Gamma)}\\\\\n&\\stackrel{\\eqref{eq:discrete invest}}\\lesssim \n\\norm{U_{\\bullet,p}-U_{\\bullet,1}}{H^{1\/2}(\\Gamma)}\n\\stackrel{\\rm (E3)}\\lesssim\n\\eta_{\\bullet,1}. \n\\end{align*}\nwhich concludes the proof.\n\n\n\\section{Numerical experiments}\n\\label{section:numerics}\n\nIn this section, we empirically investigate the performance of Algorithm~\\ref{the algorithm} on the geometries $\\Omega$ from Figure~\\ref{fig:geometries}. \nTheir boundaries $\\Gamma$ can be parametrized via rational splines of degree~$2$, i.e., there exists a $2$-open knot vector $\\widehat{\\mathcal K}_\\gamma$ on $[0,1]$, and positive weights $\\mathcal{W}_\\gamma$ such that the components of $(\\gamma_1,\\gamma_2)=\\gamma:[0,1]\\to \\Gamma$ satisfy that \n\\begin{align}\n\\gamma_1,\\gamma_2\\in\\widehat{\\mathcal{S}}^{\\,2}(\\widehat{\\mathcal K}_\\gamma,\\mathcal{W}_\\gamma);\n\\end{align}\nsee \\cite[Section~5.9]{gantner17} for details.\nOn the pacman geometry, we prescribe an exact solution $P$ of the Laplace problem as \n\\begin{equation}\nP(x_1,x_2):=r^{\\tau}\\cos\\left(\\tau\\beta\\right)\n\\end{equation}\nin polar coordinates $(x_1,x_2)=r(\\cos \\beta,\\sin \\beta)$ with $\\beta\\in(-\\pi,\\pi)$.\nSimilarly, we prescribe \n\\begin{equation}\nP(x_1,x_2):=r^{2\/3}\\cos\\left(\\frac{2}{3}(\\beta+\\pi\/2)\\right)\n\\end{equation}\nin polar coordinates $(x_1,x_2)=r(\\cos \\beta,\\sin \\beta)$ with $\\beta\\in (-3\\pi\/2,\\pi\/2)$ on the heart-shaped domain.\nFigure~\\ref{fig:solutions} shows the corresponding Dirichlet data $u=P|_\\Gamma$ as well as Neumann data $\\phi=\\partial P\/\\partial\\nu$.\nIn each case, the latter have a generic singularity at the origin. \nIt is well-known that the boundary data satisfy the hyper-singular integral equation \\eqref{eq:hyper strong} as well as the weakly-singular integral equation \\eqref{eq:weak strong}; see Appendix~\\ref{sec:weaksing} for details on the latter. \nIn the following Sections~\\ref{sec:hyper pacman}--\\ref{sec:weak heart}, we aim to numerically solve these boundary integral equations. \n\n\\begin{figure}\n\\label{fig:geometries}\n\\psfrag{Pacman}[c][c]{}\n\\psfrag{Heart}[c][c]{}\n\\centering\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/pacman.eps}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/heart.eps}\n\t\\caption{Geometries and initial nodes.}\n\\end{figure}\n\n\\begin{figure}\n\\label{fig:pacman solutions}\n\\psfrag{parameter domain}[c][c]{\\small parameter domain}\n\\psfrag{Dirichlet data pacman}[c][c]{\\small Dirichlet data $u$ on pacman}\n\\psfrag{Neumann data pacman}[c][c]{\\small Neumann data $\\phi$ on pacman}\n\\psfrag{Dirichlet data heart}[c][c]{\\small Dirichlet data $u$ on heart}\n\\psfrag{Neumann data heart}[c][c]{\\small Neumann data $\\phi$ on heart}\n\n\\centering\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/pacman_u.eps}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/pacman_phi.eps}\\\\\n\t\\hspace{2mm}\n\t\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/heart_u.eps}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/heart_phi.eps}\n\t\\caption{Dirichlet and Neumann data of the Laplace solutions $P$ from Section~\\ref{section:numerics} plotted over the parameter domain of the respective parametrization $\\gamma:[0,1]\\to\\Gamma$.}\n \\label{fig:solutions}\n\\end{figure}\n\nFor the discretization of the boundary integral equations, we employ (transformed) splines of degree $p=2$. \nBased on the knots $\\widehat{\\mathcal K}_\\gamma$ for the geometry, we choose the initial knots $\\widehat{\\mathcal K}_0$ for the discretization such that the corresponding nodes coincide, i.e., $\\widehat{\\mathcal N}_0=\\widehat {\\mathcal N}_\\gamma$.\nMoreover, we assume that all interior knots of $\\widehat{\\mathcal K}_0$ have multiplicity $1$ so that Algorithm~\\ref{the algorithm} can decide, where higher knot multiplicities are required. \nIn each case, this gives that\n\\begin{align*}\n\\widehat{\\mathcal K}_0:=\\Big(0,0,0,\\frac{1}{6},\\frac{1}{3},\\frac{1}{2},\\frac{2}{3},\\frac{5}{6},1,1,1\\Big),\\notag\\\\\n\\end{align*}\nAs basis for the considered ansatz spaces, we use \\eqref{eq:hypsing basis} for the hyper-singular equation and~\\eqref{eq:weak space} for the weakly-singular equation.\nTo (approximately) calculate the Galerkin matrix, the right-hand side vector, and the weighted-residual error estimators, we transform the singular integrands into a sum of a smooth part and a logarithmically singular part.\nThen, we use adapted Gauss quadrature to compute the resulting integrals with appropriate accuracy;\n see \\cite[Section 5]{diplomarbeit} and \\cite[Section~6]{schimanko} for details.\nFinally, we note that for the hyper-singular case, we approximate $\\phi$ by its $L^2$-orthogonal projection onto piecewise polynomials as in Section~\\ref{section:igabem}.\nWe empirically found that such an approximation is necessary for the hyper-singular equation due to stability issues of the implementation. \nWe do not apply any data approximation for the weakly-singular case. \n\nFor each example, we choose the parameters of Algorithm~\\ref{the algorithm} resp.\\ its version for the weakly-singular case of Appendix~\\ref{sec:weaksing} as $\\theta=0.5$, $\\const{min}=1$, $\\vartheta\\in\\{0,0.1,1\\}$, and $\\const{mark}=1$.\nRecall that $\\vartheta=0$ prevents any multiplicity decrease.\nFor comparison, we also consider uniform refinement with $\\theta=1$ and $\\vartheta=0$, where we mark all nodes in each step, i.e., $\\mathcal M_\\ell={\\mathcal N}_\\ell$ for all $\\ell\\in{\\mathbb N}_0$. \nNote that this leads to uniform bisection of all elements (without knot multiplicity increase).\n\n\n\\subsection{Hyper-singular integral equation on pacman}\n\\label{sec:hyper pacman}\nIn Figure~\\ref{fig:hyper pacman}, the corresponding error estimators $\\eta_\\ell$ are plotted.\nAll values are plotted in a double logarithmic scale such that the experimental convergence rates are visible as the slope of the corresponding curves.\nSince the Neumann data, which have to be resolved, lack regularity, uniform refinement regains the suboptimal rate $\\mathcal{O}(N^{-4\/7})$, whereas adaptive refinement leads to the optimal rate $\\mathcal{O}(N^{-1\/2-p})=\\mathcal{O}(N^{-5\/2})$.\nIn this example, the estimator curves look very similar for all considered $\\vartheta$.\nFor adaptive refinement, Figure~\\ref{fig:hyper pacman} additionally provides histograms of the knots $\\widehat{\\mathcal K}_\\ell$ from the last refinement step.\nMoreover, all knots with higher multiplicity than one are marked with crosses. \nNote that the exact solution $u\\circ\\gamma$ on the parameter domain (depicted in Figure~\\ref{fig:solutions}) is only $C^0$ at $1\/3$ and $2\/3$.\nWe see that $\\vartheta=0$ leads to a great amount of unnecessary multiplicity increases. \nIn contrast to this, $\\vartheta\\in\\{0.1,1\\}$ can reduce them immensely.\nIn particular, the latter choices give a much more accurate information on the regularity of the solution.\n\n\n\n\\begin{figure}\n\\label{fig:hyper pacman}\n\\psfrag{number of knots N}[c][c]{\\small number of knots $N$}\n\\psfrag{number of knots}[c][c]{\\small number of knots}\n\\psfrag{parameter domain}[c][c]{\\small parameter domain}\n\\psfrag{error estimator}[c][c]{\\small error estimator}\n\\psfrag{O(47)}[c][c]{\\tiny $\\mathcal{O}(N^{-4\/7})$}\n\\psfrag{O(52)}[c][c]{\\tiny $\\mathcal{O}(N^{-5\/2})$}\n\\psfrag{th=0.5, vth=0}[c][c]{\\small $\\theta=0.5, \\vartheta=0$}\n\\psfrag{th=0.5, vth=0.1}[c][c]{\\small $\\theta=0.5, \\vartheta=0.1$}\n\\psfrag{th=0.5, vth=1}[c][c]{\\small $\\theta=0.5, \\vartheta=1$}\n\\psfrag{ th=1, vth=0}[c][c]{\\tiny $\\theta=1, \\vartheta=0$}\n\\psfrag{ th=0.5, vth=0}[c][c]{\\tiny $\\theta=0.5, \\vartheta=0$}\n\\psfrag{ th=0.5, vth=0.1}[c][c]{\\tiny $\\theta=0.5, \\vartheta=0.1$}\n\\psfrag{ th=0.5, vth=1}[c][c]{\\tiny $\\theta=0.5, \\vartheta=1$}\n\t\\centering\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/conv_hyper_1geo_pacmanp_2}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_1geo_pacmanp_2vartheta_0}%\n\t\\\\\n\t\\hspace{2mm}\n\t\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_1geo_pacmanp_2vartheta_0.1}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_1geo_pacmanp_2vartheta_1}%\n\t\\caption{Hyper-singular integral equation on pacman (Section~\\ref{sec:hyper pacman}). Convergence plot of the error estimators $\\eta_\\ell$ and histograms of the knots $\\widehat{\\mathcal K}_\\ell$ of the last refinement step. \n\tKnots with multiplicity 3 are highlighted by a red cross, knots with multiplicity 2 by a smaller magenta cross.}\n\\end{figure}\n\n\\subsection{Weakly-singular integral equation on pacman}\n\\label{sec:weak pacman}\nIn Figure \\ref{fig:weak pacman}, the corresponding error estimators $\\eta_\\ell$ are plotted.\nSince the solution lacks regularity, uniform refinement leads to the suboptimal rate $\\mathcal{O}(N^{-4\/7})$, whereas adaptive refinement leads to the optimal rate $\\mathcal{O}(N^{-3\/2-p})=\\mathcal{O}(N^{-7\/2})$.\nFor $\\vartheta=1$, the corresponding multiplicative constant is clearly larger than for $\\vartheta\\in\\{0,0.1\\}$. \nA possible explanation is that $\\vartheta=1$ results in too few multiplicity increases. \nIndeed, the histograms in Figure~\\ref{fig:hyper pacman} of the knots $\\widehat{\\mathcal K}_\\ell$ from the last refinement step indicate that $\\vartheta\\in\\{0,0.1\\}$ leads to full multiplicity of the knots $1\/3$ and $2\/3$, which is exactly where the solution $\\phi\\circ\\gamma$ on the parameter domain (depicted in Figure~\\ref{fig:solutions}) has jumps. \nIn contrast, the choice $\\vartheta=1$ compensates the lacking regularity at these points by $h$-refinement; see also \\cite[Section~3]{resigabem}. \nAgain, $\\vartheta\\in\\{0.1,1\\}$ give a more accurate information on the regularity of the solution.\n\n\\begin{figure}\n\\label{fig:weak pacman}\n\\psfrag{number of knots N}[c][c]{\\small number of knots $N$}\n\\psfrag{number of knots}[c][c]{\\small number of knots}\n\\psfrag{parameter domain}[c][c]{\\small parameter domain}\n\\psfrag{error estimator}[c][c]{\\small error estimator}\n\\psfrag{O(47)}[c][c]{\\tiny $\\mathcal{O}(N^{-4\/7})$}\n\\psfrag{O(72)}[c][c]{\\tiny $\\mathcal{O}(N^{-7\/2})$}\n\\psfrag{th=0.5, vth=0}[c][c]{\\small $\\theta=0.5, \\vartheta=0$}\n\\psfrag{th=0.5, vth=0.1}[c][c]{\\small $\\theta=0.5, \\vartheta=0.1$}\n\\psfrag{th=0.5, vth=1}[c][c]{\\small $\\theta=0.5, \\vartheta=1$}\n\\psfrag{ th=1, vth=0}[c][c]{\\tiny $\\theta=1, \\vartheta=0$}\n\\psfrag{ th=0.5, vth=0}[c][c]{\\tiny $\\theta=0.5, \\vartheta=0$}\n\\psfrag{ th=0.5, vth=0.1}[c][c]{\\tiny $\\theta=0.5, \\vartheta=0.1$}\n\\psfrag{ th=0.5, vth=1}[c][c]{\\tiny $\\theta=0.5, \\vartheta=1$}\n\t\\centering\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/conv_hyper_0geo_pacmanp_2}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_0geo_pacmanp_2vartheta_0}%\n\t\\\\\n\t\\hspace{2mm}\n\t\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_0geo_pacmanp_2vartheta_0.1}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_0geo_pacmanp_2vartheta_1}%\n\t\\caption{Weakly-singular integral equation on pacman (Section~\\ref{sec:weak pacman}). Convergence plot of the error estimators $\\eta_\\ell$ and histograms of the knots $\\widehat{\\mathcal K}_\\ell$ of the last refinement step. \n\tKnots with multiplicity 3 are highlighted by a red cross, knots with multiplicity 2 by a smaller magenta cross.}\n\\end{figure}\n\n\n\\subsection{Hyper-singular integral equation on heart}\n\\label{sec:hyper heart}\nIn Figure~\\ref{fig:hyper heart}, the corresponding error estimators $\\eta_\\ell$ are plotted.\nSince the Neumann data, which have to be resolved, lack regularity, uniform refinement leads to the suboptimal rate $\\mathcal{O}(N^{-2\/3})$, whereas adaptive refinement leads to the optimal rate $\\mathcal{O}(N^{-1\/2-p})=\\mathcal{O}(N^{-5\/2})$.\nWhile the estimator curves look very similar, the choices $\\vartheta\\in\\{0.1,1\\}$ (allowing for knot multiplicity decrease) additionally give accurate information on the regularity of the solution; see the histograms in Figure~\\ref{fig:hyper heart}. \nNote that the (periodic extension of the) exact solution $u\\circ\\gamma$ on the parameter domain (depicted in Figure~\\ref{fig:solutions}) is only $C^0$ at $0$ resp. $1$, $1\/6$, and $5\/6$.\n\n\n\\begin{figure}\n\\label{fig:hyper heart}\n\\psfrag{number of knots N}[c][c]{\\small number of knots $N$}\n\\psfrag{number of knots}[c][c]{\\small number of knots}\n\\psfrag{parameter domain}[c][c]{\\small parameter domain}\n\\psfrag{error estimator}[c][c]{\\small error estimator}\n\\psfrag{O(23)}[c][c]{\\tiny $\\mathcal{O}(N^{-2\/3})$}\n\\psfrag{O(52)}[c][c]{\\tiny $\\mathcal{O}(N^{-5\/2})$}\n\\psfrag{th=0.5, vth=0}[c][c]{\\small $\\theta=0.5, \\vartheta=0$}\n\\psfrag{th=0.5, vth=0.1}[c][c]{\\small $\\theta=0.5, \\vartheta=0.1$}\n\\psfrag{th=0.5, vth=1}[c][c]{\\small $\\theta=0.5, \\vartheta=1$}\n\\psfrag{ th=1, vth=0}[c][c]{\\tiny $\\theta=1, \\vartheta=0$}\n\\psfrag{ th=0.5, vth=0}[c][c]{\\tiny $\\theta=0.5, \\vartheta=0$}\n\\psfrag{ th=0.5, vth=0.1}[c][c]{\\tiny $\\theta=0.5, \\vartheta=0.1$}\n\\psfrag{ th=0.5, vth=1}[c][c]{\\tiny $\\theta=0.5, \\vartheta=1$}\n\t\\centering\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/conv_hyper_1geo_heartp_2}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_1geo_heartp_2vartheta_0}%\n\t\\\\\n\t\\hspace{2mm}\n\t\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_1geo_heartp_2vartheta_0.1}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_1geo_heartp_2vartheta_1}%\n\t\\caption{Hyper-singular integral equation on heart (Section~\\ref{sec:hyper heart}). Convergence plot of the error estimators $\\eta_\\ell$ and histograms of the knots $\\widehat{\\mathcal K}_\\ell$ of the last refinement step. \n\tKnots with multiplicity 3 are highlighted by a red cross, knots with multiplicity 2 by a smaller magenta cross.}\n\\end{figure}\n\n\\subsection{Weakly-singular integral equation on heart}\n\\label{sec:weak heart}\nIn Figure \\ref{fig:weak pacman}, the corresponding error estimators $\\eta_\\ell$ are plotted.\nSince the solution lacks regularity, uniform refinement leads to the suboptimal rate $\\mathcal{O}(N^{-2\/3})$, whereas adaptive refinement leads to the optimal rate $\\mathcal{O}(N^{-3\/2-p})=\\mathcal{O}(N^{-7\/2})$.\nFigure~\\ref{fig:hyper pacman} further provides histograms of the knots $\\widehat{\\mathcal K}_\\ell$ from the last refinement step.\nOverall, we observe a similar behavior as in Section~\\ref{sec:weak pacman}.\nNote that the (periodic extension of the) exact solution $\\phi\\circ\\gamma$ on the parameter domain (depicted in Figure~\\ref{fig:solutions}) is only $C^0$ at $0$ resp. $1$, $1\/6$, and $5\/6$.\n\n\\begin{figure}\n\\label{fig:weak heart}\n\\psfrag{number of knots N}[c][c]{\\small number of knots $N$}\n\\psfrag{number of knots}[c][c]{\\small number of knots}\n\\psfrag{parameter domain}[c][c]{\\small parameter domain}\n\\psfrag{error estimator}[c][c]{\\small error estimator}\n\\psfrag{O(23)}[c][c]{\\tiny $\\mathcal{O}(N^{-2\/3})$}\n\\psfrag{O(72)}[c][c]{\\tiny $\\mathcal{O}(N^{-7\/2})$}\n\\psfrag{th=0.5, vth=0}[c][c]{\\small $\\theta=0.5, \\vartheta=0$}\n\\psfrag{th=0.5, vth=0.1}[c][c]{\\small $\\theta=0.5, \\vartheta=0.1$}\n\\psfrag{th=0.5, vth=1}[c][c]{\\small $\\theta=0.5, \\vartheta=1$}\n\\psfrag{ th=1, vth=0}[c][c]{\\tiny $\\theta=1, \\vartheta=0$}\n\\psfrag{ th=0.5, vth=0}[c][c]{\\tiny $\\theta=0.5, \\vartheta=0$}\n\\psfrag{ th=0.5, vth=0.1}[c][c]{\\tiny $\\theta=0.5, \\vartheta=0.1$}\n\\psfrag{ th=0.5, vth=1}[c][c]{\\tiny $\\theta=0.5, \\vartheta=1$}\n\t\\centering\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/conv_hyper_0geo_heartp_2}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_0geo_heartp_2vartheta_0}%\n\t\\\\\n\t\\hspace{2mm}\n\t\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_0geo_heartp_2vartheta_0.1}\\quad\n\t\\includegraphics[width=.475\\textwidth,clip=true]{figures\/hist_hyper_0geo_heartp_2vartheta_1}%\n\t\\caption{Weakly-singular integral equation on heart (Section~\\ref{sec:weak heart}). Convergence plot of the error estimators $\\eta_\\ell$ and histograms of the knots $\\widehat{\\mathcal K}_\\ell$ of the last refinement step. \n\tKnots with multiplicity 3 are highlighted by a red cross, knots with multiplicity 2 by a smaller magenta cross.}\n\\end{figure}\n\n\\section{Weakly-singular integral equation}\n\\label{sec:weaksing}%\n\n\nIn this appendix, we consider the weakly-singular integral equation\n\\begin{align}\\label{eq:weak strong}\n\\mathfrak{V}\\phi=(1\/2+\\mathfrak{K})u\\quad\\text{with given } u\\in H^{1\/2}(\\Gamma),\n\\end{align}\nwhich is equivalent to the Laplace--Dirichlet problem\n\\begin{align}\n -\\Delta P = 0 \\text{ in } \\Omega\n \\quad \\text{subject to Dirichlet boundary conditions} \\quad\n P|_\\Gamma= u \\text{ on } \\Gamma=\\partial\\Omega,\n\\end{align}\nwhere $\\phi = \\partial P\/\\partial \\nu$ is the normal derivative of the sought potential $P$.\n\n\\subsection{Functional analytic setting}\nThe weakly-singular integral equation~\\eqref{eq:weak strong} employs the single-layer operator $\\mathfrak{V}$ as well as the double-layer operator $\\mathfrak{K}'$.\nThese have the boundary integral representations\n\\begin{align}\n\\mathfrak{V}\\phi(x)=\\int_\\Gamma \\phi(y) G(x,y)\\,dy \\quad \\text{and}\\quad\n\\mathfrak{K}u(x)=\\int_\\Gamma u(y)\\frac{\\partial_y}{\\partial\\nu(y)}G(x,y)\\,dy \n\\end{align}\nfor smooth densities $\\phi,u:\\Gamma\\to{\\mathbb R}$.\n\n \nFor $0\\le \\sigma\\le1$, the single-layer operator \n${\\mathfrak V}:H^{\\sigma-1}(\\Gamma)\\to H^{\\sigma}(\\Gamma)$ and the double-layer operator $\\mathfrak{K}:H^{\\sigma}(\\Gamma)\\to H^{\\sigma}(\\Gamma)$ are well-defined, linear, and continuous. \n\nFor $\\sigma=1\/2$, \n${\\mathfrak V}:H^{-1\/2}(\\Gamma)\\to H^{1\/2}(\\Gamma)$ is symmetric and \nelliptic under the assumption that ${\\rm diam}(\\Omega)<1$, which can always be achieved by scaling $\\Omega$.\nIn particular,\n\\begin{align}\n\\edualV{\\phi}{\\psi}:=\\dual{{\\mathfrak V} \\phi}{\\psi}_\\Gamma\n\\end{align}\n defines an equivalent scalar product on \n$H^{-1\/2}(\\Gamma)$ with corresponding norm $\\enormV{\\cdot}$.\nWith this notation, the strong form~\\eqref{eq:weak strong} with data $u\\in H^{1\/2}(\\Gamma)$ is equivalently stated by\n\\begin{align}\n\\label{eq:weakform2}\n \\edualV{\\phi}{\\psi} = \\dual{(1\/2+\\mathfrak{K})u}{\\psi}_\\Gamma\n \\quad\\text{for all }\\psi\\in H^{-1\/2}(\\Gamma).\n\\end{align}\nTherefore, the Lax-Milgram lemma applies and proves that \\eqref{eq:weakform2} resp.\\ \\eqref{eq:weak strong} admits a unique solution $\\phi\\in H^{-1\/2}(\\Gamma)$.\nDetails are found, e.g., in \\cite{hw,mclean,ss,s}.\n\n\\subsection{IGABEM discretization}\nAs refinement algorithm for the boundary meshes, we use Algorithm~\\ref{alg:refinement}, where the multiplicity of the knots in step (iii) is now increased up to $p+1$ (instead of $p$), allowing for discontinuities at the nodes.\nThe set $\\mathbb{K}$ now denotes the set of all possible knot vectors that can be generated with this modified refinement algorithm starting from an initial knot vector ${\\mathcal K}_0$ as in Section~\\ref{section:boundary:discrete}, where each knot in ${\\mathcal K}_0$ might have multiplicity up to $p+1$. \nFurther, we do no longer require the restriction $w_{0,1-p}=w_{0,N_0-p}$ for the weights.\nFor ${\\mathcal K}_\\bullet\\in\\mathbb{K}$, we define the corresponding ansatz space as \n\\begin{align}\\label{eq:weak space}\n{\\mathcal X}_\\bullet:={\\mathcal{S}}^p({\\mathcal K}_\\bullet,\\mathcal{W}_\\bullet)={\\rm span}\\set{R_{\\bullet,i,p}}{i=1-p,\\dots,N_\\bullet-p}\n\\end{align}\nand the Galerkin approximation $\\Phi_\\bullet\\in{\\mathcal X}_\\bullet$ of $\\phi$ via \n\\begin{align}\n \\edualV{\\Phi_\\bullet}{\\Psi_\\bullet} = \\dual{g_\\bullet}{\\Psi_\\bullet}_\\Gamma \n\\,\\text{ for all }\\Psi_\\bullet\\in {\\mathcal X}_\\bullet,\\quad\\text{where }g_\\bullet:=(1\/2+\\mathfrak{K})u_\\bullet. \n\\end{align}\nHere, we define $u_\\bullet:=P_\\bullet\\phi$, where $P_\\bullet$ is either the identity or the Scott--Zhang operator of Section~\\ref{sec:scott zhang} onto the space of (transformed) continuous piecewise polynomials $\\mathbf{P}^p({\\mathcal{Q}}_\\bullet)\\cap C^0(\\Gamma)$.\n\n\n\nIn order to employ the weighted-residual error estimator (plus oscillations)\n\\begin{align}\n\\eta_\\bullet(z)^2:=\\norm{h_\\bullet^{1\/2}\\partial_\\Gamma(g_\\bullet-\\mathfrak{V}\\Phi_\\bullet)}{L^2(\\pi_\\bullet(z))}^2+\\norm{ h_\\bullet^{1\/2}\\partial_\\Gamma(u-u_\\bullet)}{L^2(\\pi_\\bullet(z))}^2\\quad\\text{for all }z\\in{\\mathcal N}_\\bullet, \n\\end{align}\n we require the additional regularity $u\\in H^1(\\Gamma)$. \nMoreover, we define \n\\begin{align}\n\\mu_\\bullet(z):=\\norm{ h_\\bullet^{1\/2}(1-I_{\\bullet\\ominus 1})\\Phi_\\bullet}{L^2(\\pi_\\bullet^{2p+1}(z))}\\text{ for all }z\\in{\\mathcal N}_\\bullet,\n\\end{align}\nwhere $I_\\bullet$ is now the Scott--Zhang operator onto ${\\mathcal X}_\\bullet$ defined in \\cite[Section 3.1.2]{overview}. \n\nWith these definitions, Algorithm~\\ref{the algorithm} is also well-defined for the weakly-singular case.\nAs already mentioned in Remark~\\ref{rem:feature} (b), the choice $\\vartheta=0$ and $\\mathcal M_\\ell^-=\\emptyset$ leads to no multiplicity decreases and then the adaptive algorithm coincides with the one from \\cite{resigabem} if $u_\\bullet:=u$. \nFor the latter, linear convergence at optimal rate has already been proved in our earlier work \\cite{optigabem}.\nTheorem~\\ref{thm:main} holds accordingly in the weakly-singular case and thus generalizes \\cite{optigabem}. \nWe will briefly sketch the proof in the remainder of this appendix. \n\n\n\\subsection{Reliability and efficiency}\n\\label{subsection:reliable weak}\nReliability, i.e., $\\norm{\\phi-\\Phi_\\bullet}{H^{-1\/2}(\\Gamma)}\\lesssim\\eta_\\bullet$ is already stated in \\cite[Theorem~4.4]{resigabem}. \nEfficiency, i.e., $\\eta_\\bullet\\lesssim \\norm{h_\\bullet^{1\/2}(\\phi-\\Phi_\\bullet)}{L^2(\\Gamma)}+\n\\norm{h_\\bullet^{-1\/2}(g-g_\\bullet)}{L^2(\\Gamma)}$ follows as in \\cite[Corollary~3.3]{invest}, which proves the assertion for standard BEM.\n\n\\subsection{Linear and optimal convergence}\nLinear convergence \\eqref{eq:R-linear} at optimal rate~\\eqref{eq:optimal} (with similarly defined approximability constant $\\norm{\\phi}{\\mathbb{A}_s}$) follows again from the axioms of adaptivity.\nStability (E1) follows exactly as the corresponding version \\cite[Lemma~5.1]{optigabem}. \nThe main argument is the inverse estimate $\\norm{h_\\bullet^{1\/2}\\partial_\\Gamma ({\\mathfrak V}\\Psi_\\bullet)}{L^2(\\Gamma)}\\lesssim\\norm{\\Psi_\\bullet}{H^{-1\/2}(\\Gamma)}$ of \\cite[Proposition~4.1]{optigabem} for (transformed) rational splines $\\Psi_\\bullet\\in{\\mathcal X}_\\bullet$.\nReduction (E2) follows as in \\cite[Lemma~4.4]{optigabem}.\nIt is proved via the same inverse estimate together with the contraction property \\eqref{eq:h tilde ctr} of $\\widetilde h_{\\pi_\\bullet(z)}$.\nDetails are also found in \\cite[Section~5.8.4 resp.\\ Section~5.8.5]{gantner17}.\nWe have already proved discrete reliability in \\cite[Lemma 5.2]{optigabem}; see also \\cite[Section~5.8.7]{gantner17} for details.\nThe proof of quasi-orthogonality (E4) is almost identical as for the hyper-singular case in Section~\\ref{subsec:hypsing orthogonality}. \nThe son estimate is verified as in Section~\\ref{sec:hypsing son} and the closure estimate (R2) as well as the overlay property (R3) are already found in \\cite[Proposition~2.2]{optigabem}. \n\n\\subsection{Approximability constants}\nSimilarly as in \\eqref{eq:classes}, we have that $\\norm{\\phi}{\\mathbb{A}_s^1}\\lesssim\\norm{\\phi}{\\mathbb{A}_s}\\le \n\\min\\{\\norm{\\phi}{\\mathbb{A}_s^1},\\norm{\\phi}{\\mathbb{A}_s^{p+1}}\\}\\lesssim\\norm{\\phi}{\\mathbb{A}_s^1}$, where the approximability constants are defined analogously as in Section~\\ref{sec:main}.\nThe proof follows along the lines of Section~\\ref{sec:classes}. \n\n\n\n\\section{Indirect BEM}\n\\subsection{Hyper-singular case}\nTheorem~\\ref{thm:main} remains valid if, instead of \\eqref{eq:hyper strong}, one considers the indirect integral formulation\n\\begin{align}\n\\mathfrak{W} u=\\phi\\quad\\text{with given }\\phi\\in H_0^{-1\/2}(\\Gamma),\n\\end{align}\nwhere the function $\\phi$ is again approximated via $\\phi_\\bullet$ as in Section~\\ref{section:igabem}. \nIndeed, the proof becomes even simpler due to the absence of the operator $\\mathfrak{K}'$.\n\n\\subsection{Weakly-singular case}\nSimilarly, one can consider \n\\begin{align}\n\\mathfrak{V}\\phi=u \\quad\\text{with given } u\\in H^{1\/2}(\\Gamma)\n\\end{align}\ninstead of \\eqref{eq:weak strong}, where $u$ is approximated as in Appendix~\\ref{sec:weaksing}, and the results of Appendix~\\ref{sec:weaksing} remain valid.\nAgain, the proof even simplifies due to the absence of the operator $\\mathfrak{K}$.\n\n\\section{Slit problems}\nIn contrast to before, let now $\\Gamma\\subsetneqq \\partial\\Omega$ be a connected proper subset of the boundary $\\partial\\Omega$ with parametrization $\\gamma:[a,b]\\to\\Gamma$. \nLet $E_0(\\cdot)$ denote the extension of a function on $\\Gamma$ by zero onto the whole boundary.\n\\subsection{Hyper-singular case}\nWe consider the slit problem\n\\begin{align}\n(\\mathfrak{W}E_0 u)|_\\Gamma= \\phi\\quad\\text{with given } \\phi\\in H^{-1\/2}(\\Gamma).\n\\end{align}\nHere, $H^{-1\/2}(\\Gamma)$ denotes the dual space of $\\widetilde{H}^{1\/2}(\\Gamma)=[L^2(\\Gamma),\\widetilde{H}^1(\\Gamma)]_{1\/2}$, where $\\widetilde{H}^1(\\Gamma)$ is the set of all $H^1(\\Gamma)$ functions with vanishing trace on the relative boundary $\\partial\\Gamma$.\nThe operator $(\\mathfrak{W}E_0(\\cdot))|_\\Gamma:\\widetilde{H}^{1\/2}(\\Gamma)\\to H^{-1\/2}(\\Gamma)$ is linear, continuous, and elliptic. \nLet $\\mathbb{K}$ now denote the set of all knot vectors that can be generated via Algorithm~\\ref{alg:refinement} from an initial knot vector ${\\mathcal K}_0$ on $\\Gamma$. \nFor ${\\mathcal K}_\\bullet\\in\\mathbb{K}$ and corresponding weights $\\mathcal{W}_\\bullet$ as in Section~\\ref{section:igabem} without the restriction $w_{\\bullet,1-p}=w_{\\bullet,N_\\bullet-p}$, we define the ansatz space \n\\begin{align}\n{\\mathcal X}_\\bullet:=\\set{V}{V\\circ\\gamma \\in\\widehat{\\mathcal{S}}(\\widehat{\\mathcal K}_\\bullet,\\mathcal{W}_\\bullet)\\wedge 0=V(\\gamma(a+))=V(\\gamma(b-))}\\subset \\widetilde{H}^1(\\Gamma).\n\\end{align}\nMoreover, we define the corresponding Scott--Zhang operator $J_\\bullet:L^2(\\Gamma)\\to{\\mathcal X}_\\bullet$ similarly as in Section~\\ref{sec:scott zhang} via \n\\begin{align}\nJ_\\bullet v:=\\sum_{i=2-p}^{N_\\bullet-p-1} \\Big(\\int_a^b \\widehat R_{\\bullet,i,p}^*(v\\circ\\gamma) \\,dt \\Big)R_{\\bullet,i,p}.\n\\end{align}\nReplacing the spaces $H^\\sigma(\\Gamma)$ and $H^1(\\Gamma)$ by $\\widetilde{H}^\\sigma(\\Gamma)$ and $\\widetilde{H}^1(\\Gamma)$, Proposition~\\ref{lem:Scott properties} is also satisfied in this case, where the proof additionally employs the Friedrichs inequality.\nIf we approximate $\\phi$ as in Section~\\ref{section:igabem}, Theorem~\\ref{thm:main} holds accordingly. \nThe proof follows along the same lines. \n\n\\subsection{Weakly-singular case}\nWe consider the slit problem\n\\begin{align}\\label{eq:weak slit}\n(\\mathfrak{V}E_0 \\phi)|_\\Gamma= u\\quad\\text{with}\\quad u\\in H^{1\/2}(\\Gamma),\n\\end{align}\nwhere $H^{1\/2}(\\Gamma)=[L^2(\\Gamma),H^1(\\Gamma)]_{1\/2}$.\nThe dual space of the latter is denoted by $\\widetilde{H}^{-1\/2}(\\Gamma)$.\nThe operator $(\\mathfrak{V}E_0(\\cdot))|_\\Gamma:\\widetilde{H}^{-1\/2}(\\Gamma)\\to H^{1\/2}(\\Gamma)$ is linear, continuous, and elliptic provided that ${\\rm diam}(\\Omega)<1$. \nAn adaptive IGABEM can be formulated as in Appendix~\\ref{sec:weaksing}. \nWithout multiplicity decrease and oscillation terms, it coincides with the algorithm from \\cite{resigabem}, which converges linearly at optimal rate according to \\cite{optigabem}. \nFor the generalized version, one can prove the same results as in Appendix~\\ref{sec:weaksing}. \n\n\n\n\n\n\n\\subsection*{Acknowledgement.}\nThe authors are supported by the Austrian Science Fund (FWF) through the research projects \\textit{Optimal isogeometric boundary element method} (grant P29096) and \\textit{Optimal adaptivity for BEM and FEM-BEM coupling} (grant P27005), the doctoral school \\textit{Dissipation and dispersion in nonlinear PDEs} (grant W1245), and the special research program \\textit{Taming complexity in PDE systems} (grant SFB F65).\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}~\\label{sec:intro}\n\nRadio frequencies above 100 GHz (e.g., sub-THz) are promising candidates for future communication systems to provide multi-Gbps average data rates, rapid streaming (e.g., the peak data rate on the order of Tbps) with extremely low latency (e.g., less than 1 ms), fiber-like fronthaul and backhaul in rural areas for fiber replacement and edge data centers replacements \\cite{rappaport19access,Elayan20survey,viswanathan20A, ghosh195g}. Spectrum regulators (e.g., Japanese MIAC, European CEPT, and FCC in the US) have instituted their views and provisions on frequencies above 100 GHz during the past few years, and possible coexistence and spectrum sharing techniques above 100 GHz are presented in \\cite{xing21a,marcus21a}.\n\nThe smaller wavelength (e.g., 1 mm) and wider available bandwidth (e.g., 1-40 GHz) at frequencies above 100 GHz will enable new techniques and applications like centimeter level precise position location \\cite{Kanhere20a,Kanhere19a}, wireless cognition (e.g., robotic control, human surrogate) \\cite{rappaport19access,viswanathan20A,zhang20a}, and joint communication and sensing \\cite{ali20sensing,dokhanchi19a}. Learning the characteristics of the wireless propagation channel at frequencies above 100 GHz is the initial step for researchers to design future communications systems to realize the aforementioned applications. \n\nMany universities and research centers all over the world have conducted channel sounding measurements and research at frequencies above 100 GHz, including Aalto University \\cite{nguyen2017comparing}, New York University (NYU) \\cite{xing21icc}, University of Southern California (USC) \\cite{abbasi20ICC}, Beijing Jiao Tong University (BJTU) \\cite{zhang20a}, Shanghai Jiao Tong University (SJTU) \\cite{han21icc}, Georgia Institute of Technology \\cite{Kim17a}, Brown University \\cite{ma18channel}, Technische University Braunschweig (TUBS) \\cite{priebe11channel}, Koc University \\cite{khalid16wideband}, University of Surrey \\cite{demos20a}, and Pohang University of Science and Technology \\cite{song20a}. These works have proven that sub-THz and THz frequencies can be used for future 6G wireless communications as the current mmWave 5G networks. However, what is the coverage for sub-THz communication systems and how is the system performance is still a question mark.\n\nThis paper analyzes the small-cell base station coverage and the system performance in terms of spectral efficiency (SE) at sub-THz frequencies in an Urban Microcell (UMi) environment, using realistic channel models derived from field measurements at 142 GHz conducted in downtown Brooklyn, New York \\cite{xing21icc}. Section \\ref{sec:SM} introduces the system model, channel models, and simulation settings. System performance results of single-cell and multi-cell cases for both uplink (UL) and downlink (DL) are evaluated in Section \\ref{sec:results}. Finally, concluding remarks are drawn in Section \\ref{sec:conclusion}.\n\n\n\\section{System Settings and Propagation Models}\\label{sec:SM}\nSmall-cell architecture (shrinking the cell size from a few kilometers to a few hundreds of meters) is a promising means to increase area spectral efficiency and energy efficiency in future sub-THz and THz communications \\cite{Rap17a,murdock2014consumption}. In this paper, small cell coverage is studied for both single-cell and multi-cell cases (7 cells in particular) in UMi scenarios at 142 GHz.\n\nThe propagation model and simulations in this paper are based on the outdoor wideband wireless propagation measurements at 142 GHz conducted in NYU downtown Brooklyn campus, which is a multipath-rich urban environment \\cite{xing21icc}. The measurements were conducted at sea level in clear weather and standard air condition. Fig. \\ref{fig:Mea1Loc} shows the measurement map with six TX locations and 17 RX locations (with some RX locations reused for more than one TX locations, such as RX1), resulting 16 LOS 16 LOS TX-RX location combinations and 12 NLOS TX-RX location combinations with TX-RX separation distances up to 117.4 m \\cite{xing21icc}. \n\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=0.450\\textwidth]{figures\/MeaLocation1.png}\n\t\\caption{Terrestrial urban Microcell measurement campaign in NYU's downtown Brooklyn campus. Six TX locations are identified as stars with different colors and the corresponding RX locations are identified as the same color circles \\cite{xing21icc}. }\n\t\\label{fig:Mea1Loc}\n\\end{figure}\n\nAlthough atmospheric absorption and rain attenuation beyond the natural Friis free space loss at sub-THz and THz transmissions are greater than frequencies below 6 GHz (where air causes attenuation of only fractions of dB\/km) \\cite{rappaport19access,ITU-Rattenuation}, work in \\cite{xing21a} shows that the atmospheric absorption has remarkably little impact on total path loss out to a few hundred meters and heavy rain will practically limit fixed THz links to several km. The atmospheric absorption at sea level in standard condition at 200-300 GHz is less than 10 dB\/km, and even at 800-900 GHz the additional atmospheric absorption beyond the natural Friis free space loss is 100 dB\/km at sea level, which is only 10 dB per 100 m over today's 4G cellular, which will be compensated for by the antenna gains at higher frequencies \\cite{xing21a}. \n\nIn the simulation, the base stations (BS) are set at 4 m above the ground, working as small-cell lamppost BS (the same as measurements \\cite{xing21icc}). The carrier frequency $f_c$ is at 142 GHz with 1 GHz null-to-null bandwidth, since 1 GHz to a few GHz bandwidths will be used for future sub-THz and THz communications \\cite{xing21a,rappaport19access}. The user equipment (UEs) are set at 1.5 m above the ground with 15 dBi gain antennas and a 7 dB noise figure, as shown in Table \\ref{tab:settings}.\n\n\\begin{table}[]\n\t\\caption{System parameters for uplink (UL) and downlink (DL) propagation.}~\\label{tab:settings}\n\t\\begin{tabular}{|l|l|}\n\t\t\\hline\n\t\t\\textbf{Environment} & Outdoor Urban Microcell (UMi) area \\\\ \\hline\n\t\t\\textbf{BS Height} & 4.0 m above the ground \\\\ \\hline\n\t\t\\textbf{Carrier Frequency} & DL: 142 GHz, UL: 140 GHz \\\\ \\hline\n\t\t\\textbf{DL Channel Bandwidth} & 1 GHz null-to-null \\\\ \\hline\n\t\t\\textbf{UL Channel Bandwidth} & 100 MHz null-to-null \\\\ \\hline\n\t\t\\textbf{LOS Probability Model} & NYU (squared) Model (5GCM) \\\\ \\hline\n\t\t\\textbf{Path Loss Model} & \\begin{tabular}[c]{@{}l@{}}NYU CI Path Loss Model (n, $\\sigma$):\\\\ Directional LOS: (2.1, 2.8 dB)\\\\ Directional $\\text{NLOS}_{\\text{Best}}$ (3.1, 8.3 dB)\\end{tabular} \\\\ \\hline\n\t\t\\textbf{BS Transmit Power} & 15 dBm (DL) \\\\ \\hline\n\t\t\\textbf{BS Antenna Gain} & 40 dBi \\\\ \\hline\n\t\t\\textbf{BS Noise Figure} & 5 dB \\\\ \\hline\n\t\t\\textbf{UE Transmit Power} & 0 dBm (UL) \\\\ \\hline\n\t\t\\textbf{UE Antenna Gain} & 15 dBi \\\\ \\hline\n\t\t\\textbf{UE Noise Figure} & 7 dB \\\\ \\hline\n\t\t\\textbf{Single-cell UE Distribution} & 250 UEs uniformly distributed \\\\ \\hline\n\t\t\\textbf{7-cell UE Distribution} & 1000 UEs uniformly distributed \\\\ \\hline\n\t\\end{tabular}\n\\end{table}\n\n\nThe 1 m close-in (CI) free space reference distance path loss model \\eqref{equ:CI} (with parameters derived from 142 GHz outdoor field measurements in a UMi environment in downtown Brooklyn \\cite {xing21icc, Xing21c}) is used for the link budget calculation:\n\\begin{equation}\n\t\\label{equ:CI}\n\t\\small\n\t\\begin{split}\n\t\tPL^{CI}(f_c,d_{\\text{3D}})\\;\\text{[dB]} &= \\text{FSPL}(f_c, 1 m) +10n\\log_{10}\\left( \\dfrac{d_{3D}}{d_{0}} \\right)+ \\chi_{\\sigma},\\\\\n\t\t\\text{FSPL}(f_c,1 m) &= 32.4 + 20\\log_{10}(\\dfrac{f_c}{1\\;\\text{GHz}}),\n\t\\end{split}\n\\end{equation}\nwhere FSPL$(f_c, 1 \\;\\text{m})$ is the large-scale free space path loss at carrier frequency $f_c = 142$ GHz at 1 m, $n$ is the path loss exponent (PLE), and $\\chi_{\\sigma}$ is the large-scale fading in dB (a zero mean Gaussian random variable with a standard deviation $\\sigma$ in dB). The directional PLE and shadow fading $(n, \\sigma)$ for LOS and $\\text{NLOS}_{\\text{Best}}$ directions (when the BS and UE antennas are pointing in the direction where the maximum power is received) are (2.1, 2.8 dB) and (3.1, 8.3 dB) at 142 GHz, respectively. \n\n\n\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{figures\/142GHzUMiPL2.png}\n\t\\caption{Measured and simulated directional path loss (with antenna gains removed) at 142 GHz in UMi environment. The LOS and NLOS Best path loss data (black diamonds and red crosses) are from outdoor field measurements at 142 GHz in downtown Brooklyn \\cite{xing21icc,Xing21b}, with a standard deviation of 2.8 dB and 8.3 dB to the theoretical CI model \\eqref{equ:CI} for LOS and NLOS cases, respectively. The simulated path loss data are generated by NYUSIM \\cite{NYUSIM}, with a standard deviation of 2.7 dB and 8.5 dB to the theoretical CI model \\eqref{equ:CI} for LOS and NLOS cases, respectively.}\n\t\\label{fig:PL142GHz}\n\\end{figure}\n\nThe theoretical CI path loss models and measured path loss data of outdoor field measurements at 142 GHz \\cite{xing21icc,Xing21b} in downtown Brooklyn, New York, are presented in Fig.\\ref{fig:PL142GHz}. Additional simulated path loss data are generated by NYUSIM \\cite{NYUSIM} which is an open source channel simulator based on field measurements from mmWave to sub-THz frequencies.\n\nFor radio propagation at frequencies above 100 GHz, there is still a myth that the wireless channels at higher frequencies would experience more path loss as only omni-directional antennas are considered at both the link ends \\cite{rappaport19access}. If the effective aperture is kept constant over frequencies at both the TX and RX, the path loss decrease quadratically as frequency increases, although the current antenna techniques may not be able to keep the effective aperture constant at sub-THz frequencies compared to mmWave or lower frequencies. \n\nWork in \\cite{Xing21b,Xing21c} shows that path loss exponents (PLEs) are remarkably similar over frequencies from 28 GHz to 142 GHz, when referenced to the first meter free space path loss. This means after the first meter propagation, wireless channels at sub-THz frequencies do not provide more loss to signals than the channels at mmWave frequencies. Measurements at 142 GHz have shown that metal surfaces, metal lampposts, glass walls, concrete walls, and marble pillars are good reflectors at sub-THz frequencies. Additionally, work in \\cite{Ju21Globecom,Ju21a} shows that 1-6 spatial clusters observed at 142 GHz with a 20 dB threshold form the peak for both LOS and NLOS scenarios in an UMi environment, and the number of spatial clusters follows Poisson distributions (statistics are sensitive to noise threshold). \n\n\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{figures\/SingleCell_DLSNR.png}\n\t\\caption{Single-Cell downlink coverage with SNR larger than 0 dB in UMi NLOS scenarios, assuming both BS and UE point to the $\\text{NLOS}_{\\text{Best}}$ direction.}\n\t\\label{fig:SingleCellCoverage}\n\\end{figure}\n\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{figures\/MultiCell_DLSINR.png}\n\t\\caption{7-Cell downlink coverage with SINR larger than 0 dB in UMi NLOS scenarios, assuming the UE points to the $\\text{NLOS}_{\\text{Best}}$ direction of the closest BS, and the signal powers from other BS are considered as interference.}\n\t\\label{fig:7CellCoverage}\n\\end{figure}\n\nIn our simulation, the BS EIRP (effective isotropic radiated power) is set the same as the current BS working at low mmWave frequencies (e.g., 28 GHz with an EIRP of 55 dBm), which is smaller than the maximum EIRP regulated by FCC (e.g., maximum average EIRP of 60 dBm for BS and 40 dBm for mobile). The BS is set at the origin in the single-cell case with a radius of 200 m coverage for a signal to noise ratio (SNR) larger than 0 dB in UMi NLOS scenarios. Fig. \\ref{fig:SingleCellCoverage} shows the single-cell downlink coverage calculated using \\eqref{equ:CI} with PLE $n = 3.1$ for $\\text{NLOS}_{\\text{Best}}$ scenarios (average path loss without fading). A single base station with an EIRP of 55 dBm at 142 GHz can cover an area with a radius of 200 m even in NLOS scenarios.\n\nFor the multi-cell case (7-cell in particular), a BS is set at the origin with six other BS equally spaced, as shown in Fig. \\ref{fig:7CellCoverage}. Considering NLOS scenarios, the UEs point to the $\\text{NLOS}_{\\text{Best}}$ direction where the maximum power is received, and the signal powers from other BS are considered as interference. It shows that 7-cell architecture can provide a coverage area (with SINR larger than 0 dB) with a radius of 400 m in NLOS scenarios. The LOS coverage areas of both single-cell and 7-cell architectures are much larger than in NLOS environments. Thus, to study the coverage and system performance at 142 GHz, 1000 UEs are uniformly distributed in areas with a radius of 200 m and 400 m for single-cell and 7-cell cases, respectively.\n\nThe NYU (squared) LOS probability model \\eqref{equ:LOS} (a statistical model derived from a real-world database in downtown New York City \\cite{A5GCM15,Rap17a}) is used to predict whether the UE is within a clear LOS of the BS or in an NLOS region due to obstructions:\n\\begin{equation}\n\t\\label{equ:LOS}\n\t\\begin{split}\n\tP_{LOS} (d_{2D}) = & (\\min(d_1\/d_{2D}, 1)(1-\\exp(-d_{2D}\/d_2)) \\\\\n\t& +\\exp(-d_{2D}\/d_2))^2,\n\t\\end{split}\n\\end{equation}\nwhere $P_{LOS} (d_{2D})$ is the likelihood that a UE is in a clear LOS of the BS, $d_{2D}$ is the 2D Euclidean separation distance between the BS and UE in meters, $d_1 = 22$ m, and $d_2 = 100$ m. \n\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{figures\/UElocations3.png}\n\t\\caption{UE distributions for single-cell and 7-cell scenarios.}\n\t\\label{fig:UElocations}\n\\end{figure}\n\nFig. \\ref{fig:UElocations} shows the distribution of UE for both single-cell and 7-cell scenarios, and different colors of UEs representing the SNR level with dark blue corresponding to outage (with SNR less than -10 dB). The LOS status of each UE is determined by \\eqref{equ:LOS} and about 20\\% of the UE is in LOS of the BS in the single-cell case, as shown in Fig. \\ref{fig:UElocations}(a). It is worth noting that, in the 7-cell case, a UE is determined the LOS status of each BS based on the BS-UE separation distance using \\eqref{equ:LOS}, and about 37\\% of UEs are in LOS of at least one BS, as shown in Fig. \\ref{fig:UElocations}(b). \n\nIn the single-cell case, the UEs in LOS of the BS point to the LOS boresight directions to the BS and the UEs in NLOS region point to the $\\text{NLOS}_{\\text{Best}}$ directions. The downlink (DL) received power is calculated by: \n\\begin{equation}\n\t\\label{equ:SNR}\n\t\\begin{split}\n\t\tPr_{dl} = Pt_{bs} + G_{bs} + G_{ue} - \tPL^{CI}(f_c,d_{\\text{3D}}),\\\\\n\t\\end{split}\n\\end{equation} \nwhere the PLE and shadow fading of (2.1, 2.8 dB) and (3.1, 8.3 dB) are used for directional path loss models in LOS and $\\text{NLOS}_{\\text{Best}}$ scenarios. As shown in Fig. \\ref{fig:UElocations}(a), there are ~10\\% of the UEs are out of coverage (with an SNR lower than 0 dB).\n\nIn the multi-cell case, the UEs point to the directions where the maximum powers are received (i.e., the LOS boresight directions or the strongest $\\text{NLOS}_{\\text{Best}}$ directions to a certain BS). The signal powers from the rest six BS are considered as interference. As shown in Fig. \\ref{fig:UElocations}(b), there are ~15\\% of the UEs are out of coverage (with an SINR lower than 0 dB).\n\nIt is worth noting that the propagation model used in this paper are based on field measurements conducted in an UMi environment at sea level in a clear weather and standard air condition, and additional effects (e.g., rain\/snow\/fog, different altitudes and humidity) can be added to the models for different scenarios.\n\n\\section{Simulation Results of Single Cell and Multicell Cases}\\label{sec:results}\n\nThe spectral efficiency (SE) of data transmission is an important performance of communication systems, which usually varies across the service and depends on the UE locations as well as channel conditions \\cite{xing21vtc}. The user SE is defined as the amount of data transferred to (DL) or from (UL) the user normalized by the time-frequency resources used by the system for data transfer. Suppose $M_{u}$ packets have been transmitted for user $u$, the SE $\\eta_u$ is calculated as:\n\\begin{equation}\n\t\\eta_u = \\left( \\sum_{i=1}^{M_u}N_{u,i} \\right)\/ \\left( \\sum_{i=1}^{M_u}T_{u,i}B_{u,i}\\right), \n\\end{equation}\nwhere $N_{u,i}, T_{u,i}, B_{u,i}$ are respectively the number of information bits, transmission time, and allocated bandwidth for the $i$-th packet of user $u$.\n\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{figures\/SEDL.png}\n\t\\caption{Simulated downlink User Spectral Efficiency at 142 GHz in UMi environment with a downlink channel bandwidth of 1 GHz.}\n\t\\label{fig:SEDL}\n\\end{figure}\n\nFig. \\ref{fig:SEDL} presents the downlink spectral efficiency (SE) distribution at 142 GHz for both the single-cell case (with a radius of 200 m) and 7-cell case (with a radius of 400 m) with the BS and UEs using directional antennas, as shown in Table \\ref{tab:settings}. In the single-cell case (blue link in Fig. \\ref{fig:SEDL}), the average user SE is 3.22 bps\/Hz, the median user SE is 1.89 bps\/Hz, and the cell-edge user (5\\% of the users either at the edge of the cells or in deep fading area) SE is 0.15 bps\/Hz. With a 1 GHz bandwidth, the average downlink data rate is 3.22 Gbps and the cell-edge users experience an average downlink date of 150 Mbps. The 7-cell architecture can increase the total coverage to a radius of 400 m (by adding six more BS and shrinking the cell size of each BS), with the average user SE of 4.46 bps\/Hz, the median user SE of 3.80 bps\/Hz, and the cell-edge user SE of 0.04 bps\/Hz. With a 1 GHz bandwidth, the average downlink data rate for the 7-cell case is 4.46 Gbps and the cell-edge users experience an average downlink date of 40 Mbps.\n\nIt shows that using the 7-cell architecture can greatly improve the average (+40\\%) and median user SE (+100\\%) compared to the single-cell architecture, however, the cell-edge users experience worse service (-75\\%) due to the interference. \n\nFig. \\ref{fig:SEUL} shows the uplink user spectral efficiency distribution at 142 GHz for both the single-cell case (with a radius of 200 m) and the 7-cell case (with a radius of 400 m) with the BS and UEs using directional antennas. In the single-cell case, the uplink SE decrease significantly compared to the downlink SEs, with the mean SE of 2.39 bps\/Hz (-26\\%), the median SE of 1.93 bps\/Hz (-50\\%), and the cell-edge SE of 0.02 bps\/Hz (-87\\%). The uplink coverage is limited by the low EIRP of the UE devices (15 dBm). With a 100 MHz bandwidth, the average data rate in the sing-cell case is 239 Mbps and the cell-edge data rate is 20 Mbps.\n\nIn the 7-cell case, the uplink SEs also decrease (compared to the downlink SEs) due to the limited UE EIRP, with the mean SE of 3.25 bps\/Hz (-27\\%), the median SE of 1.93 bps\/Hz (-49\\%), and the cell-edge SE of 0.02 bps\/Hz (-50\\%). In general, using the 7-cell architecture can greatly improve the average and median SE compared to using the single-cell architecture, but have a worse performance for the cell-edge users. \n\n\n\\begin{figure} \n\t\\centering\n\t\\includegraphics[width=0.45\\textwidth]{figures\/SEUL.png}\n\t\\caption{Simulated uplink user spectral efficiency at 140 GHz in UMi environment with an uplink channel bandwidth of 100 MHz.}\n\t\\label{fig:SEUL}\n\\end{figure}\n\n\\section{Conclusion}\\label{sec:conclusion}\nIn this paper, we have analyzed both the single-cell and multi-cell base station coverage of sub-THz systems at 142 GHz in outdoor UMi scenarios based on realistic channel models derived from field measurements. The single-cell base station can provide coverage of a 200 m radius, and the 7-cell base stations can provide total coverage of a 400 m radius. Uplink and downlink system performances in terms of spectral efficiency are studied for both single-cell and multi-cell cases. The 7-cell architecture can greatly improve the coverage and system performance in terms of the average and median SE compared to the single-cell case but has a worse performance to the cell-edge users. In the 7-cell case, with a 1 GHz bandwidth, DL sector throughput is 4.5 Gbps and DL cell-edge throughput is 40 Mbps. The uplink coverage is limited by UE transmit power and UE antenna gain. Dynamic blockages (e.g., human and vehicle blockages) and fast beam switching (when there is an obstruction block the link) algorithms will be studied for future work. \n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nEmpirically, quasars are remarkable for spectral energy distributions\n(SEDs) with substantial power spanning many decades of frequency from\nthe radio through the hard X-ray \\citep[e.g.,][]{ElvisEtal1994}. This SED\nencodes (though perhaps not unambiguously) \nthe fundamental physical properties of the accreting black\nhole system such as the mass, spin, and accretion rate as well as\nrandom effects due to orientation; determining the mapping from the\nphysical properties of interest to their spectral signatures is the\nultimate goal of quasar SED studies.\n\nThe accretion flow near the black hole generates the vast majority of\nthe continuum emission from the near-infrared through the hard X-rays,\nwhile infrared continuum light longward of $\\sim1$\\,\\micron\\ is\ngenerally attributed to a parsec-scale, cold, and dusty region that is\nheated by and reprocesses the higher energy, direct continuum and\nserves to obscure the inner accretion system and broad-line region in\ntype~2 (here defined as narrow-line) quasars. While the community\ngenerically refers to this medium as a `torus', this convenient\npicture merely accounts for the empirical evidence for axisymmetric\ndust obscuration without providing a coherent physical description of\nthis structure (see \\citealt{KoKa1994} and \\citealt{ElSh2006} for\npossible dynamical models). Regardless of the specific model, the\nquasar torus might be considered the interface between a quasar and\nits host galaxy, a reservoir for gas to fuel the black hole and\npotentially a site of star formation. Understanding the nature of the\ntorus is a primary aim of infrared quasar studies.\n\n\\citet[][hereafter R06]{ric+06} recently presented 24\\,\\micron\\ through\n$u$-band photometric data of the largest sample to date of type~1\n(broad-line) quasars. The resulting 259 SEDs, supplemented with\narchival VLA, {\\em GALEX}, and {\\em ROSAT} data as available, span a\nbolometric luminosity range of approximately three decades. Overall, SEDs\nof type~1 quasars are remarkably similar out to $z=6$\n\\citep[e.g.,][]{ohad+06,hines+06,jiang+06}. However, in a qualitative\ncomparison of two composite SEDs made from higher and lower luminosity\nobjects, R06 noted subtle differences indicating that luminosity\naffected overall SED shapes, particularly in the mid-infrared\n(1.3--8\\,\\micron) as shown in Figure~\\ref{fig:sed_zoom}.\nSuch differences in the mid-infrared spectra of quasars with higher\nmid-infrared fluxes were noted previously in the SEDs of a smaller sample of\n35 optically selected quasars by \\citet{hatz+05}, and even earlier in\nan SED study of 29 Seyfert galaxies \\citep{EdMa1986}.\n\nLuminosity is already known to affect some quasar emission properties.\nFor example, the Baldwin Effect characterizes the decreasing\nequivalent widths of ultraviolet emission lines with increasing\nultraviolet continuum luminosity\n\\citep[e.g.,][]{baldwin,ric+02,shang+03}. In addition, the maximum\nvelocity of outflows seen in \\ion{C}{4}\\ absorption increases with\nultraviolet luminosity \\citep{LaoBra2002}. Across broader spectral\nregions, the X-ray-to-optical flux ratio is known to decrease with\nincreasing ultraviolet luminosity\n\\citep[e.g.,][]{AvTa1986,Strateva+2005}. Finally, and perhaps most\nrelevant for the current study, the ratio of type~1 to type~2 quasars\nis found to increase with luminosity, as found in X-ray\n\\citep[e.g.,][]{ueda+03,akylas+06} and narrow emission-line surveys\n\\citep{hao+05}. Overall, these empirical relationships indicate that\nluminosity affects the quasar system in some fashion.\n\nIn this paper, we investigate potential SED differences of type~1\nquasars as a function of luminosity, focusing in particular on the\nrest-frame mid-infrared and optical\/UV emission with the goal of\nunderstanding how the quasar affects its immediate environment.\nThroughout we assume $\\Lambda$-CDM cosmology with $\\Omega_{\\rm\nM}=0.3$, $\\Omega_{\\Lambda}=0.7$, and $H_0=70$\\mbox{\\,km\\,s$^{-1}$}\\,Mpc$^{-1}$\n\\citep{sperg03,sperg06}.\n\n\\section{Analysis and Results}\n\\label{sec:analysis}\n\nWe take as the starting point for our analysis the photometric SED\ndata from Tables 1 and 2 of R06, focusing on the spectral regions with\ncomplete coverage: the observed-frame optical and mid-infrared. The\noptical coverage is from the Sloan Digital Sky Survey $ugriz$\nphotometric data \\citep{sdss_ref} corrected for Galactic extinction.\nThe mid-infrared data are from {\\em Spitzer} IRAC 3.6, 4.5, 5.8, and\n8.0\\micron\\ (full widths at half maxima [FWHM] of the spectral\nresponses are 0.7, 1.0, 1.4, and 2.9\\micron, respectively) and MIPS\n\\mbox{24\\micron} (FWHM=5.3\\micron) imaging. There has\nbeen no attempt to remove host galaxy light, which is likely to\ncontribute most significantly to the SED at rest-frame $\\sim1$\\micron.\n\nR06 included all SDSS quasars with publicly available {\\emph{Spitzer}}\\\ncoverage as of the Data Release 3 quasar catalog of \\citet{dr3_qso}.\nThe SDSS is quite complete to all but the reddest quasars to $i=19.1$\n\\citep{ric+06b}; all SDSS quasars were detected by {\\emph{Spitzer}}. From the\nentire R06 sample of 259 quasars, those that are known to be radio\nluminous ($\\log(L_{\\rm R}) > 32$ erg\\mbox{\\,s$^{-1}$}~Hz$^{-1}$ at rest-frame\n20~cm assuming $\\alpha_{\\rm \\nu,R}=-1$) or have no 24\\,\\micron\\\nphotometry have been eliminated. Note that the majority of R06\nquasars (229\/259) have only radio upper limits. Of these 229, 99 do\nnot have sensitive enough flux limits to rule out high radio\nluminosity. The trimmed sample comprises 234\\ quasars with\n$z=0.14$--5.21, $\\bar{z}=1.54\\pm0.92$, $i=16.8$--20.4,\n$\\bar{i}=18.9\\pm0.7$, $M_{i}=$--22.0 to --28.2, and\n$\\bar{M_{i}}=-25.6\\pm1.6$.\n\nThroughout this section, when we refer to a linear fit, we are using\nthe ordinary least-squares bisector method to determine the slope and\nintercept of a line fitted to (usually logarithmic) data values.\n\\citet{isobe+90} recommend this method for determining the functional\nrelationship between ordinate and abscissa values when the errors are\nnot well defined or insignificant. Errors quoted for the slopes are\nthe standard deviations of the best-fitting values.\n\n\\subsection{Relative Integrated Infrared and Optical Luminosities}\n\\label{sec:lopt}\n\nAs a first step, we investigated whether the relative power in the\ninfrared vs. the optical varies with luminosity. As mid-infrared\nemission and optical photons are generated in physically distinct\nregions, there is no reason a priori why the relative power should (or\nshould not) be affected by luminosity. Figure~\\ref{fig:lumplots}a\nshows the logarithm of the ratio of infrared and optical luminosities\nvs. the logarithm of the optical luminosity; the quantity $l_{\\rm ir}$--$l_{\\rm opt}$\\\nmay be considered a parameterization of the reprocessed to direct\naccretion-powered luminosity. An alternate depiction of the same\ndata, $l_{\\rm ir}$\\ vs. $l_{\\rm opt}$\\ is presented in Figure~\\ref{fig:lumplots}b. We\nuse $l_{\\rm ir}$\\ and $l_{\\rm opt}$\\ to indicate $\\log(L_{\\rm opt})$ and $\\log(L_{\\rm\nir})$, respectively, with units of \\mbox{\\,erg~s$^{-1}$}; Both values are taken from\nTable~1 of R06. The first, $l_{\\rm ir}$, is the luminosity integrated from\n$\\log(\\nu)=12.48$--14.48 Hz (1--100\\micron); $l_{\\rm opt}$\\ is integrated from\n$\\log(\\nu)=14.48$--15.48 Hz (0.1--1\\micron). Power-law interpolation\nwas used on the monochromatic luminosities calculated from the\ntabulated flux densities.\n\nFrom visual inspection, there is a clear trend with\nincreasing optical luminosity for quasars to have relatively more\npower in the optical, and a linear fit gives a slope of $-0.51\\pm0.06$\n(1$\\sigma$ uncertainty). Non-parametric statistical tests of the\nbivariate data measuring Spearman's $\\rho$=--0.288 and Kendall's\n$\\tau$=--0.194 indicate that the probability of obtaining these values\nif the parameters are not correlated is $<1\\times10^{-5}$; the\nstatistical values and probabilities are listed in\nTable~\\ref{tab:stats}.\n\nGiven that $l_{\\rm opt}$\\ characterizes direct emission from the accretion\ndisk, it may be considered in some sense a truer measure of overall\nquasar power than the indirect and reprocessed infrared light. A\nstraightforward interpretation of this significant trend would be that\na smaller fraction of the accretion-powered luminosity is reprocessed\nand emitted in the mid-infrared in more luminous quasars. However,\noptical photons are subject to dust reddening and extinction. The\nintegrated optical luminosity measured by the observer would be\nreduced by extinction while $l_{\\rm ir}$--$l_{\\rm opt}$\\ would increase in the same\nsense as the correlation. Therefore, if dust extinction is prevalent,\nit could induce the correlation seen in Figure~\\ref{fig:lumplots}a.\nAs the small grains responsible for moderate optical extinction will\nnot affect the infrared power (and may not in any case be along the\nsame line of sight), $l_{\\rm ir}$\\ may be a more robust measure of the total\nquasar power. Indeed, we perform the same statistical tests on\n$l_{\\rm ir}$--$l_{\\rm opt}$\\ vs. $l_{\\rm ir}$\\ and find that these two quantities are not\ncorrelated. The scatter about the mean is also roughly the same across\nalmost three decades in $l_{\\rm ir}$.\n\n\\subsection{Comparison of Optically Blue and Red Quasars}\n\\label{sec:reddening}\n\nThe values of $l_{\\rm ir}$--$l_{\\rm opt}$\\ evident in Figure~\\ref{fig:lumplots}a span\nroughly one decade from --0.5 to 0.5. As the apparent trend of\n$l_{\\rm ir}$--$l_{\\rm opt}$\\ vs. $l_{\\rm opt}$\\ plotted in Figure~\\ref{fig:lumplots}a may be\ninduced by extinction, we investigate if extinction could also\ncontribute to the width of the spread in $l_{\\rm ir}$--$l_{\\rm opt}$. To simply\nparameterize the color of the optical continuum, a power-law model\n($l_\\nu\\propto\\nu^{\\alpha_{\\nu{\\rm, opt}}}$) was fit to the SDSS\nphotometric data between rest-frame 1200--5000\\,\\AA. This region of\nthe spectrum is typically dominated by the quasar, and by inspection\nof the continuum fits compared to the data (2--5 points in this\nregion); a power-law model is appropriate for this purpose. For\nillustration, three sample SEDs are shown in Figure~\\ref{fig:sample}.\nFrom analysis of the SDSS composite quasar spectrum, the host galaxy\ncontribution reddens the optical continuum slope for\n$\\lambda>5000$\\,\\AA\\\n\\citep{VandenBerk2001}. Wavelengths $<1200$\\,\\AA\\ are to be avoided\nas the intervening Ly$\\alpha$ forest reduces the flux significantly.\n\nA histogram of $\\alpha_{\\nu{\\rm,opt}}$\\ values is shown in Figure~\\ref{fig:hists}a. At\nthe blue (more positive) end, the distribution is roughly gaussian; a\nlong tail extends to the red (negative) end. The qualitative shape of\nthis $\\alpha_{\\nu{\\rm,opt}}$\\ distribution is consistent with that of $\\Delta(g-i)$, the SDSS\n$g-i$ color minus the mode of the $g-i$ color of quasars at the same\nredshift \\citep{ric+03}. The very red tail in the $\\Delta(g-i)$\\ distribution\nis associated with continuum dust reddening, though intrinsically red\nquasars may contribute somewhat. To illustrate the effect of optical\nextinction on Figures~\\ref{fig:lumplots}a, a reddening vector has been\ndrawn. The vector reflects the decrease in $l_{\\rm opt}$\\ of $A_{\\rm g}=0.1$\nmagnitudes of extinction, assuming Small Magellanic Cloud (SMC)\nreddening (as appropriate for quasars; e.g., \\citealt{hopkins+04}) of\nan\n$\\alpha_{\\nu{\\rm,opt}}$$=-0.49$ optical continuum integrated from 1000--10,000\\,\\AA\\\n(following R06). \n\nWe use the median $\\alpha_{\\nu{\\rm,opt}}$=$-0.49$ to divide the sample into `optically\nred' ($\\alpha_{\\nu{\\rm,opt}}$\\,$<-0.49$) and `optically blue' ($\\alpha_{\\nu{\\rm,opt}}$\\,$\\ge-0.49$)\ngroups. A comparison of the histograms of\n$l_{\\rm ir}$--$l_{\\rm opt}$\\ for the groups (Figure~\\ref{fig:hists}b) shows an\noffset. A student's t-test of the two populations indicates that the\nprobability that they have the same mean is $<1\\times10^{-6}$. This\nmean reduction in $l_{\\rm ir}$--$l_{\\rm opt}$\\ for the red quasars is consistent with\nthe expectation from extinction. In fact, within the optically red\nquasars, the subset of clearly dust-reddened ($\\alpha_{\\nu{\\rm,opt}}$$<-0.70$) quasars\nshow the most extreme values of $l_{\\rm ir}$--$l_{\\rm opt}$.\n\nTherefore, we conclude that optical extinction makes a significant\ncontribution to the width in the $l_{\\rm ir}$--$l_{\\rm opt}$\\ distribution, and likely\ninduces the correlation of $l_{\\rm ir}$--$l_{\\rm opt}$\\ vs. $l_{\\rm opt}$. In fact, this\ncorrelation disappears when only blue quasars are considered (see\nTable~\\ref{tab:stats}). For optically blue quasars, arguably the\nleast likely to suffer from dust reddening, the data are consistent with a\nconstant ratio of infrared to optical luminosity over approximately\nthree decades in luminosity: $L_{\\rm ir}\\propto L_{\\rm opt}$ (see\nFigure~\\ref{fig:lumplots}b).\n\n\\subsection{Dependence of the Mid-Infrared SED on Luminosity}\n\\label{sec:midir_spec}\n\nA close-up of the composite SEDs from R06 in the spectral region from\n1--12\\,\\micron\\ reveals distinctions between the luminous and faint\nquasar SEDs (Figure~\\ref{fig:sed_zoom}). When the SEDs are all\nnormalized to 1.3\\,\\micron, the luminous SED shows enhanced flux in\nthe mid-infrared spectral region relative to the mean and faint\ncomposites. The three SEDs then almost meet again for\n$\\lambda\\gtrsim$10\\micron. Note that these composites were\nconstructed from broad-band photometry, and therefore do not reflect\ninteresting mid-infrared features such as strong 10\\micron\\ silicate\nemission recently seen in several quasar spectra\n\\citep[e.g.,][]{hao+05b,sieb+05}.\n\nTo investigate if these differences are evident trends in all of the\ndata rather than just in the composites, power-law fits have been\nperformed to the {\\emph{Spitzer}}\\ data in the rest-frame 1.8--8\\,\\micron\\\nregion. As available, 15\\micron\\ {\\em ISO} and $JHK$ photometry were\nalso incorporated. This spectral region was chosen to maximize the\nphotometric coverage while reducing to the extent possible\ncontamination from the host galaxy (which peaks in the near-infrared)\nand silicate emission (at $\\sim 10$\\,\\micron). Even in logarithmic\nunits, curvature is evident in this spectral region\n(Figure~\\ref{fig:sed_zoom}), and so the spectral index $\\alpha_{\\nu{\\rm,ir}}$\\\n($l_\\nu\\propto\\nu^{\\alpha_{\\nu{\\rm, ir}}}$) should only be considered\na simple parameterization of the data. Sample SEDs and the\ncorresponding fits are shown in Figure~\\ref{fig:sample}. The value\nof $\\alpha_{\\nu{\\rm,ir}}$\\ for eight quasars with $z>3.5$ could not be measured because\nthe MIPS 24\\,\\micron\\ data point was the only one in the bandpass.\n\nA histogram of the distribution of $\\alpha_{\\nu{\\rm,ir}}$\\ values is shown in\nFigure~\\ref{fig:hists}a. For reference, the mean and standard\ndeviation of the $\\alpha_{\\nu{\\rm,ir}}$\\ values, $-1.20\\pm0.34$, are consistent with the\n1--10\\micron\\ spectral indices measured by \\citet{haas+03} of\n$-1.3\\pm0.3$ for 64 Palomar-Green quasars observed with {\\em ISO}.\n\nAs anticipated from the composite spectral results, the distribution\nof $\\alpha_{\\nu{\\rm,ir}}$\\ vs. $l_{\\rm ir}$\\ shows a clear trend with increasing $l_{\\rm ir}$\\ whereby\nthe more luminous quasars tend to have steeper (more negative)\nspectral indices in the mid-infrared; this is evident in\nFigure~\\ref{fig:air}. This apparent correlation is supported with the\nbivariate Spearman and Kendall tests, which give probabilities that\n$\\alpha_{\\nu{\\rm,ir}}$\\ is not correlated with $l_{\\rm ir}$\\ of $6.9\\times10^{-5}$ and\n$2.7\\times10^{-5}$, respectively. (We use $l_{\\rm ir}$\\ instead of $l_{\\rm opt}$\\ to\navoid the problems of underestimating the luminosity due to\nextinction.)\n\nNext, we investigate if quasars are also brighter at 1.8--8\\,\\micron\\\nrelative to the optical continuum region. We choose to measure the\nmonochromatic luminosities at 3\\,\\micron\\ and 5000\\,\\AA\\ ($l_{3\\mu{\\rm m}}$\\ and\n$l_{5000{\\rm \\AA}}$\\ in logarithmic units) to represent the mid-infrared and\noptical, respectively. Three \\micron\\ is near the middle of the\n1.8--8.0\\,\\micron\\ bandpass, has good photometric coverage, and should\nhave minimal contribution from even a star-forming host galaxy;\n5000\\,\\AA\\ marks the red end of the quasar-dominated optical\/UV\ncontinuum region. As seen from the results in Table~\\ref{tab:stats},\n$l_{3\\mu{\\rm m}}$--$l_{5000{\\rm \\AA}}$ does not change significantly as $l_{3\\mu{\\rm m}}$\\ increases.\nIf only optically blue quasars are considered (to reduce the potential\neffects of extinction), there is still no significant evidence for any\nchange in $l_{3\\mu{\\rm m}}$--$l_{5000{\\rm \\AA}}$\\ for increasing $l_{3\\mu{\\rm m}}$. The steepening\nof the mid-infrared continuum evident from $\\alpha_{\\nu{\\rm,ir}}$\\ vs. $l_{\\rm ir}$\\ is much\nclearer evidence of mid-infrared changes.\n\nIn the course of the previous tests, we visually examined each of the\nbest-fitting optical and mid-infrared power-law fits in comparison\nwith all of the SDSS and {\\emph{Spitzer}}\\ photometry (cf.,\nFigure~{\\ref{fig:sample}). Perhaps surprisingly, up to $z\\sim1.2$,\nextrapolating a simple power-law fit from the IRAC photometry predicts\nquite well $L_{\\nu}$ in the MIPS 24\\micron\\ bandpass. From $z=1.2$--2,\nhowever, the extrapolation typically {\\em overpredicts} $L_{\\nu}$ in\nthe MIPS 24\\micron\\ bandpass. (Beyond $z=2$, the MIPS datapoint is\nincluded in the power-law fit.) This systematic effect is not because\nof a contribution of significant 10\\micron\\ flux from silicate\nemission, which would cause the opposite effect. Instead, the deficit\nof 24\\micron\\ emission is evidence for curvature in the mid-infrared\nspectrum. This is consistent with the luminous and faint composite\nSEDs, which when normalized at 1.3\\micron\\ almost meet again near\n10\\micron\\ (see Fig.~\\ref{fig:sed_zoom} and R06).\n\nTo characterize spectral curvature, we calculate the monochromatic\n8\\micron\\ luminosity, $L_{8\\mu{\\rm m}}$, in two ways. For the first,\n$L_{8,\\alpha}$, the best-fitting mid-infared power-law spectral index and\nnormalization are used to extrapolate to rest-frame 8\\micron. For the\nsecond, $L_{\\rm 8,SED}$, photometric datapoints within 0.3 dex of\nrest-frame 8\\micron\\ are used to normalize the R06 composite SED to\nmeasure the 8\\micron\\ luminosity. The lowest panel in\nFigure~\\ref{fig:sample} illustrates the discrepancy between\n$L_{\\rm 8,SED}$\\ and $L_{8,\\alpha}$\\ seen in some objects. Quasars without at\nleast three data points in the 1.8--8,0\\micron\\ regime and photometry\nwithin 0.3 dex of 8\\micron\\ were eliminated from this analysis to\nmitigate redshift effects, the resulting sample has 146 objects. As\nseen in Figure~\\ref{fig:leight}, the strong decrease in\n$L_{\\rm 8,SED}$\/$L_{8,\\alpha}$\\ with increasing $L_{\\rm ir}$\\ is indicative of\nsignificant curvature in the spectrum (see Table~\\ref{tab:stats}).\n\nOverall, we find strong evidence for significant differences in the\nmid-infrared spectral regime as a function of luminosity.\nSpecifically, more luminous quasars show steeper (more negative) values of $\\alpha_{\\nu{\\rm,ir}}$.\nExtrapolating $\\alpha_{\\nu{\\rm,ir}}$\\ to rest-frame 8\\micron\\ also overpredicts the\n8\\micron\\ luminosity in luminous quasars, thus demonstrating \nspectral curvature. Both these results are consistent with the\nexpectation from the luminous composite shown in\nFigure~\\ref{fig:sed_zoom}.\n\n\\subsection{Investigating Potential Systematic Redshift Effects}\n\\label{sec:z_effects}\n\nQuasar spectral studies have historically been plagued with\ndifficulties in distinguishing between luminosity and redshift effects\n(see \\citealt{steffen+06} for a recent example) particularly as\nspectral redshifting causes distinct spectral regions to be sampled\nin a given observed-frame bandpass. \n\nIn this study, the evident mid-infrared spectral curvature and the gap\nin the {\\emph{Spitzer}}\\ photometric coverage between observed-frame 8 and\n24\\micron\\ could induce apparent systematic differences in the\nmid-infrared spectra as a result of redshift. To examine this\npotential problem, we focused on a narrow redshift slice in our\nsample, and reran the statistical tests of $\\alpha_{\\nu{\\rm,ir}}$\\ vs. $l_{\\rm ir}$. We choose\nthe redshift range from $z=1.0$--1.5 because there is still a\nsignificant ($\\gtrsim1$~dex) infrared luminosity range and at least 3\nIRAC bands sample the rest-frame wavelength range of interest. This\nredshift subsample comprises 55\\ quasars. Even for this\nsignificantly smaller sample, the correlation of $\\alpha_{\\nu{\\rm,ir}}$\\ with $l_{\\rm ir}$\\\nremains statistically significant, with both Spearman's and Kendall's\ntests giving $<1\\times10^{-4}$ probabilities of no correlation.\nFurthermore, if the effect were entirely induced by bandpass effects,\n$\\alpha_{\\nu{\\rm,ir}}$\\ would be expected to correlate significantly with redshift,\nwhich is not the case (see Table~\\ref{tab:stats}). More even\nphotometric sampling is likely needed to better constrain the shape of\nthe correlation and certainly to investigate spectral curvature in\ngreater detail.\n\n\\subsection{Comparing Optical and Infrared Spectral Indices}\n\nIn the standard understanding of the infrared emission in quasars,\ncold, dusty material on parsec scales from the supermassive black hole\nis illuminated by and reprocesses accretion disk radiation. The torus\ngeometry is typically described as some form of flattened disk-like\nconfiguration, encompassing toroids, open cones, and flared disks. At\nmid-infrared (1--10\\micron) wavelengths, the continuum spectrum is\ndominated by thermal dust emission, and as long as the highest dust\ntemperature is fixed (by dust sublimation, for example), the SED is\nnot sensitive to the spectral shape of the illuminating continuum\nregardless of the assumed geometry \\citep{IvEl1997}. Only at\nnear-infrared ($<3$\\micron) wavelengths is the illuminating spectral\nshape expected to affect the SED as a result of scattering of direct\ncontinuum photons (Nenkova et al., in prep.). As seen in\nFigure~\\ref{fig:alphas}, from these data we find no evidence for any\ncorrelation between $\\alpha_{\\nu{\\rm,opt}}$\\ and $\\alpha_{\\nu{\\rm,ir}}$, confirming theoretical\nexpectation. Even when considering just the optically blue quasars to\nreduce the potential masking effects of optical extinction, the two\nparameters show no relation. Therefore, there is no evidence that the\nmid-infrared spectral changes are induced by the shape of the\nilluminating continuum; this is also consistent with the lack of\nnotable differences between the composite SEDs constructed by R06 from\noptically red and optically blue quasars.\n\n\n\\subsection{Evaluating Mid-Infrared and Optical Bolometric Corrections}\n\nTypically, quasars only have photometric data in a few bandpasses\nwhile the quantity of interest (e.g., for estimating black hole masses\nand Eddington accretion rates) is the integrated bolometric\nluminosity. Therefore, finding a single point in the SED from which a\nrobust estimate of the bolometric luminosity can be made would be\nuseful. Given the evidence discussed in \\S\\ref{sec:lopt} for the\npotentially significant effects of dust extinction on $l_{\\rm opt}$, we\ninvestigate whether a mid-infrared point would be more robust. Using\nthe 3\\micron\\ monochromatic luminosity, $\\nu L_{\\rm 3 \\mu m}$, we\ncalculated both the overall (to $L_{\\rm bol}$) bolometric corrections\n(BCs) and the infrared (to $L_{\\rm ir}$) BCs. The value of $L_{\\rm\nbol}$ is integrated from 100\\micron\\ to 10~keV; where photometry do\nnot exist to measure this quantity, the composite SED was used to fill\nout the wavelength coverage. These are also compared to the typical,\n5100\\AA\\ BCs as compiled in R06. Histograms of the three BC\ndistributions are shown in Figure~\\ref{fig:bc}. The 3\\micron\\ BC,\n8.59$\\pm$3.30 is notably quite comparable to the 5100\\AA\\ BC,\n10.47$\\pm$4.14, with fractional standard deviations of 38\\%\\ and 40\\%,\nrespectively. The extremely tight and symmetric 3\\micron\\ infrared BC\nindicates that $\\nu L_{\\rm 3 \\mu m}$ is quite a good proxy for $L_{\\rm\nir}$.\n\n\\section{Discussion}\n\\label{sec:dis}\n\nThe distribution of $l_{\\rm ir}$\\ vs. $l_{\\rm opt}$\\ as plotted in\nFigure~\\ref{fig:lumplots}b is notably quite tight. As expected in a\nparadigm where the direct optical\/UV continuum emission powers thermal\nemission in the infrared, optical and infrared luminosity increase in\ntandem. While the overall sample of 234 quasars gives $L_{\\rm\nir}\\propto L^{0.94\\pm0.02}$, considering only the blue quasars (the\nleast likely to be dust-extincted) gives a linear relationship. This\nconstant ratio of optical to infrared luminosities over $\\sim3$\ndecades in luminosity suggests that the dust mass (estimated from the\ninfrared luminosity) radiating in the regime probed by {\\emph{Spitzer}}\\ also\nincreases approximately linearly with optical luminosity (assuming\nsimilar grain properties and distributions). At the same time, the\ninner radius of the dust-emitting region increases as $\\sqrt{L_{\\rm\nUV}}$ as the dust sublimation radius moves out. Thus the inner wall\nof the torus at any luminosity is expected to have a similar\ntemperature. It is challenging to extrapolate this point to\ninferences about the total infared emitting volume, as for optically\nthick emission (expected for $\\lambda<10$\\micron), the observed\nluminosity does not depend only on volume, but also on, for example,\ndust covering fraction, self-shadowing, and viewing angle. Therefore,\nit is not clear how our results are related to the interpretation that\ntorus geometry must change with quasar power, as understood from the\nincreasing type~1\/type~2 ratio with luminosity found empirically in\nX-ray \\citep[][]{ueda+03,akylas+06} and narrow emission-line surveys\n\\citep{hao+05}.\n\nIndications that quasar luminosity is affecting the torus structure\nare the significant spectral changes in the mid-infrared.\nSpecifically, we seen an enhancement of mid-infrared emission in more\nluminous quasars manifested as a steepening of the 1.8--8.0\\micron\\\ncontinuum. As expected from the composite SEDs, this enhancement is\nbetter described as a bump, as an extrapolation from $\\alpha_{\\nu{\\rm,ir}}$\\\noverpredicts the luminosity at $\\sim8$\\micron\\ in luminous quasars.\nIn current static torus models, the shape of the mid-infrared\ncontinuum shortward of the 10\\micron\\ silicate feature is sensitive to\nnumerous physical inputs, including (1) orientation effects\n\\citep[e.g.,][]{nenkova+02}, (2) the dust grain properties\n\\citep[e.g., see Fig. 6 of][]{schart+05}, (3) the clumpiness and\nnumber of emitting clouds (\\citealt{DuVanB2005}; Nenkova et al., in\nprep.), and (4) the opening angle of the torus \\citep[e.g., see\nFig. 20 of][]{fritz+06}. Our focus on type~1 quasars has reduced the\neffect of (1) to the extent possible. Similarly, given that the inner\nwall of the torus is set by the graphite sublimation temperature to be\n$T\\sim1500$~K, a luminosity dependence of dust grain size,\ncomposition, and distribution is not expected, though detailed\ninfrared spectroscopy is required to examine this issue empirically.\nRegarding (3), current clumpy torus models do not generate explicit\npredictions for any luminosity dependence of the mid-infrared\ncontinuum. Therefore, option (4) seems plausible given the other\nevidence for a luminosity-dependent covering fraction of the torus.\nHowever, we are not aware of explicit predictions in the literature\nfor how a changing opening angle would affect the mid-infrared\ncontinuum that are consistent with our results.\n\nAnother possible explanation for the enhanced mid-infrared spectra of\nluminous quasars is that a hotter dust component may be contributing\nmore significantly to their SEDs than in lower luminosity objects. A\nstriking example of this type of component was recently presented by\n\\citet{RodMaz06} in the near-infrared spectrum of the narrow-line\nSeyfert 1 galaxy, Mrk 1239; they interpreted it as thermal emission\nfrom $T\\sim1200$~K graphite grains. This type of spectral feature,\nthe so-called `near-infrared bump', has been noted for years\n\\citep[e.g.,][]{HyAl1982}, and its physical origin was interpreted in\nthe same fashion by \\citet{barvainis+87}. Similar structure is also\nevident in the mean radio-quiet composite spectrum of\n\\nocite{ElvisEtal1994} Elvis et al. (1994; though not, interestingly,\nin the radio-loud composite --- see their Fig.~10), while the more\nrecent combined SEDs of \\citet{hatz+05} also show some hint of the\nnear-infared bump (see their Fig.~9). In fact, \\citet{EdMa1986} (who\nmodeled the feature as a parabola peaked at 5.2\\micron\\ in\n$\\log(f_{\\nu})$ vs. $\\log(\\nu)$ units) found its strength to be\ncorrelated with luminosity in their study of 29 quasar SEDs. Given\nthe IRAC wavelength sampling, at higher redshifts, the data are\nprobing the shorter wavelengths of the mid-infrared bandpass, and the\npresence of a `hot bump' peaking between 3--5\\micron\\ would manifest\nitself as a steepening in $\\alpha_{\\nu{\\rm,ir}}$. The overprediction of $L_{8\\mu{\\rm m}}$\\ from\nextrapolations from the mid-infrared power law further supports this\ninterpretation. This conclusion is not necessarily inconsistent with\na `receding torus' paradigm whereby more luminous quasars have tori\ncovering less of the sky \\citep{lawrence91} as a flatter dusty\nstructure would be less likely to block lines of sight to the hottest\ndust.\n\nFinally, we also mention that polyaromatic hydrocarbon (PAH) emission\nis a possible contributor to the mid-infrared enhancement in more\nluminous quasars. We consider this unlikely however, as the strongest\nPAH features (at 6.2\\micron\\ and 7.7\\micron) are not sampled by the\nIRAC data beyond $z\\sim0.3$ and do not enter the MIPS 24\\micron\\\nbandpass until $z\\sim3$. Other PAH features in the 2--5\\micron\\ range\nare much weaker in star-forming regions and laboratory spectra\n\\citep{peeters+04}, though they have not been well-studied in luminous\nquasar spectra.\n\nThe gross similarity of quasar mid-infrared through optical SEDs is\nconsistent with the finding of \\citet{brown+06} that both mid-infrared\nand optical type~1 quasar selection give similar space densities and\nluminosity functions for $z=1$--5. Therefore, though optical dust\nextinction may induce a false correlation as in\nFigure~\\ref{fig:lumplots}a, it is unlikely to distort strongly the\nstudy of type~1 quasars. Similar to our findings, a lack of variation\nof the ratio of far-infrared to optical luminosity as a function of\nbolometric luminosity was found by \\citet{andreani+99} for a sample of\n120 quasars (with many far-infrared upper limits). However, we\ncaution on the use of far-infrared ($\\gtrsim20$\\micron) luminosities\nto characterize quasars as star formation in the host galaxy can\ncontribute significantly to or even dominate the power in this regime.\nFor example, from the 1--100\\micron\\ spectral modeling of 36 type~1\nquasars with luminosities matched to our sample ($L_{\\rm quasar} >\n3\\times10^{44}$\\mbox{\\,erg~s$^{-1}$}) presented in \\citet{fritz+06}, the range of\nmodeled quasar contributions to the total 5--1000\\micron\\ luminosities\nranged from 9--100\\%\\ with a mean of $62\\pm26\\%$. The correlation\nbetween the far-infrared luminosities and 7.7\\micron\\ PAH emission\nseen by \\citet{schweitz+06} in a sample of quasars and ultra-luminous\ninfrared galaxies further supports star-formation as a significant\n(and variable) source of far-infrared power in luminous active\ngalaxies. However, in typical torus+star formation model fits, the\nquasar dominates from $\\sim2$ to 10\\micron\\\n\\citep[e.g.,][]{vanBDu2003}, even considering some contribution from\nPAH emission. As the integrated $L_{\\rm ir}$\\ values used in this work were\nderived from IRAC+24\\micron\\ MIPS data, they may more accurately\nrepresent the quasar power than infrared luminosities estimated from\nlonger wavelength data.\n\n\\section{Summary and Conclusions}\n\nWe have analyzed 234\\ radio-quiet quasar SEDs from the sample of\nR06 with the goal of investigating possible mid-infrared SED changes\nas a function of luminosity. While we find quasar SEDs overall to be\nremarkably stable with luminosity in this wavelength regime, there are\nsome notable differences. We detail the following results:\n\n\\begin{enumerate}\n\n\\item{Over three decades in infrared luminosity, the mean ratio and\n scatter of $L_{\\rm ir}$\/$L_{\\rm opt}$\\ vs. $L_{\\rm ir}$\\ in quasar SEDs is constant. Some\n part of the spread in the distribution of $L_{\\rm ir}$\/$L_{\\rm opt}$\\ is incurred\n from optical dust extinction. Given that dust extinction can\n significantly reduce $L_{\\rm opt}$, the mid-infrared luminosity may be a\n more robust indicator of bolometric luminosity.}\n\n\\item{The steepening of the 1.8--8.0\\,\\micron\\ spectral index, $\\alpha_{\\nu{\\rm,ir}}$,\n with luminosity coupled with the spectral curvature\n evident from $L_{8\\mu{\\rm m}}$\\ measurements indicate that more luminous\n quasars show a mid-infrared enhancement consistent with a\n 3--5\\micron\\ bump. This result confirms the qualitative differences\n noted by R06 between luminous and faint composite quasar SEDs. As\n noted in previous studies, this feature can be attributed to an\n increasing contribution from hot ($T>1000$\\,K) dust with increasing\n quasar power \\citep{barvainis+87}.}\n\n\\item{The overall bolometric correction from the monochromatic 3\\micron\\\n luminosity, 8.59$\\pm$3.30, has a comparable dispersion to the\n typical bolometric correction from $\\nu L_{\\rm 5100\\AA}$ without\n suffering from dust extinction. The infrared bolometric correction\n from $\\nu L_{\\rm 3\\mu m}$ to $L_{\\rm ir}$, $3.44\\pm1.68$, shows a very tight and\n symmetric distribution, and is therefore a good measure of\n mid-infrared power.}\n\\end{enumerate}\n\nThough the first two results point strongly to a change in the\ngeometry of the mid-infrared emitting region with luminosity, we know\nof no predictions from current torus models that are consistent with\nthese findings. The notable 1.8--8.0\\micron\\ spectral changes we have found\ntherefore furnish strong constraints for future theoretical modeling\nof mid-infrared torus emission.\n\nStatic models for the torus are ultimately unsatisfying; our results\nsupport the picture where quasars are shaping the environment that\nemits in the mid-infrared -- the `torus' is not simply a passive\nreprocessor of UV\/optical emission. The dynamical wind\nparadigm, where magnetohydrodynamic and radiation pressure lift and\naccelerate a dusty wind away from the central engine is therefore more\nappealing \\citep{KoKa1994,EvKoAr2002,ElSh2006}, and radiation pressure naturally\nhas a luminosity dependence. Detailed predictions of mid-infrared\nspectra within this framework are a necessary next step, and future\nstudies of 1--10\\micron\\ infrared spectra of type~1 quasars matched in\nredshift with a range of luminosities will illuminate the nature of\nthe spectral differences we find. Comparisons with theoretical models of such data will\nenable constraints on physical properties of interest such as the\ntemperature, structure, and radius of the dusty region to aid in\nunderstanding the role of luminosity in shaping the quasar\nenvironment.\n\n\\acknowledgements \nWe thank Maia Nenkova, Matt Malkan, and Bev Wills for sharing their expertise.\nSupport for SCG was provided by NASA through the {\\em Spitzer}\nFellowship Program, under award 1256317. \nThis research has made use of \nthe Sloan Digital Sky Survey (http:\/\/www.sdss.org). \n\n{\\it Facilities:} {Spitzer ()}, {Sloan ()} \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}