Adaptive boundary element methods with convergence rates

This paper presents adaptive boundary element methods for positive, negative, as well as zero order operator equations, together with proofs that they converge at certain rates. The convergence rates are quasi-optimal in a certain sense under mild assumptions that are analogous to what is typically assumed in the theory of adaptive finite element methods. In particular, no saturation-type assumption is used. The main ingredients of the proof that constitute new findings are some results on a posteriori error estimates for boundary element methods, and an inverse-type inequality involving boundary integral operators on locally refined finite element spaces.

[1]  Wolfgang Dahmen,et al.  Element-by-Element Construction of Wavelets Satisfying Stability and Moment Conditions , 1999, SIAM J. Numer. Anal..

[2]  Carsten Carstensen,et al.  Convergence of adaptive boundary element methods , 2012 .

[3]  L. R. Scott,et al.  Finite element interpolation of nonsmooth functions satisfying boundary conditions , 1990 .

[4]  V. Girault,et al.  A Local Regularization Operator for Triangular and Quadrilateral Finite Elements , 1998 .

[5]  Birgit Faermann,et al.  Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations , 1998 .

[6]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[7]  C. Schwab,et al.  Boundary Element Methods , 2010 .

[8]  Wolfgang Dahmen,et al.  Adaptive methods for boundary integral equations: Complexity and convergence estimates , 2007, Math. Comput..

[9]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods for General Second Order Linear Elliptic PDEs , 2005, SIAM J. Numer. Anal..

[10]  B Faermann.  Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary elements methods. Part I. The two-dimensional case , 2000 .

[11]  Doyoon Kim,et al.  Trace theorems for Sobolev-Slobodeckij spaces with or without weights , 2007 .

[12]  Ricardo H. Nochetto,et al.  A posteriori error analysis for a class of integral equations and variational inequalities , 2010, Numerische Mathematik.

[13]  Martin Costabel,et al.  Boundary Integral Operators on Lipschitz Domains: Elementary Results , 1988 .

[14]  Karsten Urban.  Adaptive Wavelet Methods , 2008 .

[15]  Rob P. Stevenson,et al.  Computation of Singular Integral Operators in Wavelet Coordinates , 2005, Computing.

[16]  G. Verchota.  Layer potentials and regularity for the Dirichlet problem for Laplace's equation in Lipschitz domains , 1984 .

[17]  Ricardo H. Nochetto,et al.  Convergence of Adaptive Finite Element Methods , 2002, SIAM Rev..

[18]  Christian Kreuzer,et al.  Quasi-Optimal Convergence Rate for an Adaptive Finite Element Method , 2008, SIAM J. Numer. Anal..

[19]  H. Johnen,et al.  On the equivalence of the K-functional and moduli of continuity and some applications , 1976, Constructive Theory of Functions of Several Variables.

[20]  W. Dörfler.  A convergent adaptive algorithm for Poisson's equation , 1996 .

[21]  Michael Karkulik,et al.  Efficiency and Optimality of Some Weighted-Residual Error Estimator for Adaptive 2D Boundary Element Methods , 2013, Comput. Methods Appl. Math..

[22]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Asymptotically Optimal Complexity Estimates , 2006, SIAM J. Numer. Anal..

[23]  Zhonghai Ding,et al.  A proof of the trace theorem of Sobolev spaces on Lipschitz domains , 1996 .

[24]  Rob P. Stevenson,et al.  Optimality of a Standard Adaptive Finite Element Method , 2007, Found. Comput. Math..

[25]  Stefan A. Funken,et al.  Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D , 2013 .

[26]  Ricardo H. Nochetto,et al.  Data Oscillation and Convergence of Adaptive FEM , 2000, SIAM J. Numer. Anal..

[27]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods II—Beyond the Elliptic Case , 2002, Found. Comput. Math..

[28]  Birgit Faermann,et al.  Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods Part II. The three-dimensional case , 2002, Numerische Mathematik.

[29]  Carsten Carstensen,et al.  Mathematical foundation of a posteriori error estimates and adaptive mesh-refining algorithms for boundary integral equations of the first kind , 2001 .

[30]  Dirk Praetorius,et al.  Estimator reduction and convergence of adaptive BEM , 2012, Applied numerical mathematics : transactions of IMACS.

[31]  P. Clément.  Approximation by finite element functions using local regularization , 1975 .

[32]  Wolfgang Dahmen,et al.  Adaptive Finite Element Methods with convergence rates , 2004, Numerische Mathematik.

[33]  Peter Oswald,et al.  Multilevel Finite Element Approximation , 1994 .

[34]  Wolfgang Dahmen,et al.  Wavelets on Manifolds I: Construction and Domain Decomposition , 1999, SIAM J. Math. Anal..

[35]  Rob Stevenson,et al.  On the Compressibility of Operators in Wavelet Coordinates , 2004, SIAM J. Math. Anal..

[36]  Carsten Carstensen,et al.  Residual-based a posteriori error estimate for hypersingular equation on surfaces , 2004, Numerische Mathematik.

[37]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[38]  Rob P. Stevenson,et al.  An optimal adaptive wavelet method without coarsening of the iterands , 2006, Math. Comput..

[39]  Carsten Carstensen,et al.  Numerische Mathematik A posteriori error estimate and h-adaptive algorithm on surfaces for Symm ’ s integral equation , 2001 .

[40]  Wolfgang Dahmen,et al.  Inverse inequalities on non-quasi-uniform meshes and application to the mortar element method , 2003, Math. Comput..

[41]  ROB STEVENSON,et al.  The completion of locally refined simplicial partitions created by bisection , 2008, Math. Comput..

[42]  Rob Stevenson,et al.  Locally Supported, Piecewise Polynomial Biorthogonal Wavelets on Nonuniform Meshes , 2000 .

[43]  Christoph Ortner,et al.  Convergence of simple adaptive Galerkin schemes based on h − h/2 error estimators , 2010, Numerische Mathematik.

[44]  J. Marschall,et al.  The trace of Sobolev-Slobodeckij spaces on Lipschitz domains , 1987 .

[45]  Reinhold Schneider,et al.  Multiskalen- und Wavelet-Matrixkompression , 1998 .

[46]  Michael Karkulik,et al.  Residual A-Posteriori Error Estimates in BEM : Convergence of h-Adaptive Algorithms , 2011 .

[47]  Tsogtgerel Gantumur,et al.  An optimal adaptive wavelet method for nonsymmetric and indefinite elliptic problems , 2008 .

[48]  Michael Karkulik,et al.  Quasi-optimal Convergence Rate for an Adaptive Boundary Element Method , 2013, SIAM J. Numer. Anal..

[49]  Mark Ainsworth,et al.  The Conditioning of Boundary Element Equations on Locally Refined Meshes and Preconditioning by Diagonal Scaling , 1999 .

[50]  Stefan A. Funken,et al.  Energy norm based a posteriori error estimation for boundary element methods in two dimensions , 2009 .