Convergence of Adaptive BEM and Adaptive FEM-BEM Coupling for Estimators Without h-Weighting Factor

Abstract We analyze adaptive mesh-refining algorithms in the frame of boundary element methods (BEM) and the coupling of finite elements and boundary elements (FEM-BEM). Adaptivity is driven by the two-level error estimator proposed by Ernst P. Stephan, Norbert Heuer, and coworkers in the frame of BEM and FEM-BEM or by the residual error estimator introduced by Birgit Faermann for BEM for weakly-singular integral equations. We prove that in either case the usual adaptive algorithm drives the associated error estimator to zero. Emphasis is put on the fact that the error estimators considered are not even globally equivalent to weighted-residual error estimators for which recently convergence with quasi-optimal algebraic rates has been derived.

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

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

[3]  Michael Karkulik,et al.  Quasi-optimal convergence rates for adaptive boundary element methods with data approximation, part I: weakly-singular integral equation , 2014 .

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

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

[6]  Kunibert G. Siebert,et al.  A convergence proof for adaptive finite elements without lower bound , 2011 .

[7]  Michael Karkulik,et al.  Inverse estimates for elliptic boundary integral operators and their application to the adaptive coupling of FEM and BEM , 2012, 1211.4360.

[8]  Carsten Carstensen,et al.  Averaging Techniques for the Effective Numerical Solution of Symm's Integral Equation of the First Kind , 2005, SIAM J. Sci. Comput..

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

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

[11]  H. Yserentant.  On the multi-level splitting of finite element spaces , 1986 .

[12]  Carsten Carstensen,et al.  Axioms of adaptivity , 2013, Comput. Math. Appl..

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

[14]  Norbert Heuer,et al.  An adaptive boundary element method for the exterior Stokes problem in three dimensions , 2006 .

[15]  Tsogtgerel Gantumur,et al.  Adaptive boundary element methods with convergence rates , 2011, Numerische Mathematik.

[16]  Kunibert G. Siebert,et al.  A BASIC CONVERGENCE RESULT FOR CONFORMING ADAPTIVE FINITE ELEMENTS , 2008 .

[17]  E. Zeidler.  Nonlinear Functional Analysis and Its Applications: II/ A: Linear Monotone Operators , 1989 .

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

[19]  Olaf Steinbach,et al.  Is the one‐equation coupling of finite and boundary element methods always stable? , 2013 .

[20]  D. Praetorius,et al.  A posteriori error estimates for the Johnson–Nédélec FEM–BEM coupling , 2012, Engineering analysis with boundary elements.

[21]  Norbert Heuer,et al.  Adaptive Boundary Element Methods , 2014 .

[22]  Ernst P. Stephan,et al.  Two-level methods for the single layer potential in ℝ3 , 1998, Computing.

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

[24]  Carsten Carstensen,et al.  Adaptive coupling of boundary elements and finite elements , 1995 .

[26]  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.

[27]  Norbert Heuer,et al.  A posteriori error estimators, adaptivity, convergence, and implementation , 2014 .

[28]  B. M. Fulk.  MATH , 1992 .

[29]  Bernd Eggers,et al.  Nonlinear Functional Analysis And Its Applications , 2016 .

[30]  Ernst P. Stephan,et al.  An Adaptive Two-Level Method for the Coupling of Nonlinear FEM-BEM Equations , 1999 .

[31]  W. Hackbusch,et al.  Finite elements on degenerate meshes: inverse-type inequalities and applications , 2005 .

[32]  N. Heuer.  On the equivalence of fractional-order Sobolev semi-norms , 2012, 1211.0340.

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

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

[35]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

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

[37]  Franco Brezzi,et al.  On the coupling of boundary integral and finite element methods , 1979 .

[38]  Michael Karkulik,et al.  Energy norm based error estimators for adaptive BEM for hypersingular integral equations , 2015 .

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

[40]  S. Singh.  Nonlinear functional analysis and its applications , 1986 .

[41]  Francisco-Javier Sayas,et al.  The Validity of Johnson-Nédélec's BEM-FEM Coupling on Polygonal Interfaces , 2009, SIAM Rev..

[42]  Jens Markus Melenk,et al.  Institute for Analysis and Scientific Computing , 2015 .

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

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

[45]  Michael Karkulik,et al.  Classical FEM-BEM coupling methods: nonlinearities, well-posedness, and adaptivity , 2012, 1211.4225.

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

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

[48]  Ernst P. Stephan,et al.  An adaptive two-level method for hypersingular integral equations in \(R^3\) , 2000 .

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

[50]  Jacobo Bielak,et al.  An exterior interface problem in two-dimensional elastodynamics , 1983 .

[51]  Michael Vogelius,et al.  Feedback and adaptive finite element solution of one-dimensional boundary value problems , 1984 .

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

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

[54]  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 .

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

[56]  E. Stephan,et al.  Coupling of FEM and BEM for a transmission problem with nonlinear interface conditions. Hierarchical and residual error indicators , 2012 .

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

[58]  Michael Feischl,et al.  Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems , 2012 .

[59]  L. Evans.  Measure theory and fine properties of functions , 1992 .

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

[61]  H. Yserentant.  Erratum. On the Multi-Level Splitting of Finite Element Spaces.(Numer. Math. 49, 379-412 (1986)). , 1986 .