Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation

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• 作者：Michael Feischl (1)
  Thomas Führer (1)
  Michael Karkulik (2)
  Jens Markus Melenk (1)
  Dirk Praetorius (1)
  
• 关键词：Boundary element method ; Weakly ; singular integral equation ; A posteriori error estimate ; Adaptive algorithm ; Convergence ; Optimality ; 65N30 ; 65N38 ; 65N50 ; 65R20 ; 41A25
• 刊名：Calcolo
• 出版年：2014
• 出版时间：December 2014
• 年：2014
• 卷：51
• 期：4
• 页码：531-562
• 全文大小：475 KB
• 参考文献：1. Aurada, M., Feischl, M., Führer, T., Karkulik, M., Praetorius, D.: Efficiency and optimality of some weighted-residual error estimator for adaptive 2D boundary element methods. Comput. Methods Appl. Math. 13, 305-32 (2013)
  2. Aurada, M., Feischl, M., Führer, T., Melenk, J., Praetorius, D.: Inverse estimates for elliptic boundary integral operators and their application to the adaptive coupling of FEM and BEM. ASC Report 07/2012. Institute for Analysis and Scientific Computing, Vienna University of Technology (2012)
  3. Aurada, M., Feischl, M., Kemetmüller, J., Page, M., Praetorius, D.: Each \(H^{1/2}\) -stable projection yields convergence and quasi-optimality of adaptive FEM with inhomogeneous Dirichlet data in \({\mathbb{R}}^d\) . Math. Model. Numer. Anal. 47, 1207-235 (2013)
  4. Aurada, M., Ferraz-Leite, S., Goldenits, P., Karkulik, M., Mayr, M., Praetorius, D.: Convergence of adaptive BEM for some mixed boundary value problem. Appl. Numer. Math. 62(4), 226-45 (2012) CrossRef
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  9. Carstensen, C., Praetorius, D.: Averaging techniques for the a posteriori BEM error control for a hypersingular integral equation in two dimensions. SIAM J. Sci. Comput. 29(2), 782-10 (2007) (electronic)
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  14. Feischl, M., Führer, T., Praetorius, D.: Adaptive FEM with optimal convergence rates for a certain class of non-symmetric and possibly non-linear problems. ASC Report 43/2012. Institute for Analysis and Scientific Computing, Vienna University of Technology (2012)
  15. Feischl, M., Karkulik, M., Melenk, J., Praetorius, D.: Quasi-optimal convergence rate for an adaptive boundary element method. SIAM J. Numer. Anal. 51, 1327-348 (2013) CrossRef
• 作者单位：Michael Feischl (1)
  Thomas Führer (1)
  Michael Karkulik (2)
  Jens Markus Melenk (1)
  Dirk Praetorius (1)
  
  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstra?e 8-10, 1040?, Vienna, Austria
  2. Departamento de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicu?a Mackenna, 4860, Santiago, Chile
  
• ISSN：1126-5434

文摘

We analyze an adaptive boundary element method for Symm’s integral equation in 2D and 3D which incorporates the approximation of the Dirichlet data \(g\) into the adaptive scheme. We prove quasi-optimal convergence rates for any \(H^{1/2}\) -stable projection used for data approximation.