Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D

A posteriori error estimation is an important tool for reliable and efficient Galerkin boundary element computations. For hypersingular integral equations in 2D with a positive-order Sobolev space, we analyse the mathematical relation between the (h − h/2)-error estimator from [S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method, Computing 83 (2008), pp. 135–162], the two-level error estimator from [M. Maischak, P. Mund, and E. Stephan, Adaptive multilevel BEM for acoustic scattering, 585 Comput. Methods Appl. Mech. Eng. 150 (1997), pp. 351–367], and the averaging error estimator from [C. Carstensen and D. Praetorius, Averaging techniques for the a posteriori bem error control for a hypersingular integral equation in two dimensions, SIAM J. Sci. Comput. 29 (2007), pp. 782–810]. All of these a posteriori error estimators are simple in the following sense: first, the numerical analysis can be done within the same mathematical framework, namely localization techniques for the energy norm. Second, there is almost no implementational overhead for the realization.

[1]  M. A. Jaswon.  Boundary Integral Equations , 1984 .

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

[3]  Carsten Carstensen,et al.  Averaging Techniques for a Posteriori Error Control in Finite Element and Boundary Element Analysis , 2007 .

[4]  Ricardo H. Nochetto,et al.  Small data oscillation implies the saturation assumption , 2002, Numerische Mathematik.

[5]  Carsten Carstensen,et al.  Averaging Techniques for the A Posteriori BEM Error Control for a Hypersingular Integral Equation in Two Dimensions , 2007, SIAM J. Sci. Comput..

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

[7]  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 .

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

[9]  Carsten Carstensen,et al.  Adaptive Boundary Element Methods for Some First Kind Integral Equations , 1996 .

[10]  Norbert Heuer,et al.  hp-adaptive Two-Level Methods for Boundary Integral Equations on Curves , 2001, Computing.

[11]  Dirk Praetorius,et al.  Simple a posteriori error estimators for the h-version of the boundary element method , 2008, Computing.

[12]  Dirk Praetorius,et al.  Estimator Reduction and Convergence of Adaptive FEM and BEM , 2009 .

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

[14]  Carsten Carstensen,et al.  A posteriori error estimates for hp--boundary element methods , 1996 .

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

[16]  Carsten Carstensen,et al.  A posteriori error estimates for boundary element methods , 1995 .

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

[18]  Carsten Carstensen,et al.  An a posteriori error estimate for a first-kind integral equation , 1997, Math. Comput..

[19]  Carsten Carstensen,et al.  Efficiency of a posteriori BEM-error estimates for first-kind integral equations on quasi-uniform meshes , 1996, Math. Comput..

[20]  Olaf Steinbach,et al.  Numerical Approximation Methods for Elliptic Boundary Value Problems: Finite and Boundary Elements , 2007 .

[21]  Ernst P. Stephan,et al.  Adaptive multilevel BEM for acoustic scattering , 1997 .

[22]  Norbert Heuer.  An hp-adaptive refinement strategy for hypersingular operators on surfaces , 2002 .

[23]  R. Hiptmair,et al.  Boundary Element Methods , 2021, Oberwolfach Reports.

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

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