Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1147-1173.

We consider the symmetric FEM-BEM coupling for the numerical solution of a (nonlinear) interface problem for the 2D Laplacian. We introduce some new a posteriori error estimators based on the (h - h/2)-error estimation strategy. In particular, these include the approximation error for the boundary data, which allows to work with discrete boundary integral operators only. Using the concept of estimator reduction, we prove that the proposed adaptive algorithm is convergent in the sense that it drives the underlying error estimator to zero. Numerical experiments underline the reliability and efficiency of the considered adaptive mesh-refinement.

DOI : 10.1051/m2an/2011075
Classification : 65N30, 65N15, 65N38
Mots clés : FEM-BEM coupling, a posteriori error estimate, adaptive algorithm, convergence
@article{M2AN_2012__46_5_1147_0,
     author = {Aurada, Markus and Feischl, Michael and Praetorius, Dirk},
     title = {Convergence of some adaptive {FEM-BEM} coupling for elliptic but possibly nonlinear interface problems},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {1147--1173},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {5},
     year = {2012},
     doi = {10.1051/m2an/2011075},
     mrnumber = {2916376},
     zbl = {1276.65066},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an/2011075/}
}
TY  - JOUR
AU  - Aurada, Markus
AU  - Feischl, Michael
AU  - Praetorius, Dirk
TI  - Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2012
SP  - 1147
EP  - 1173
VL  - 46
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an/2011075/
DO  - 10.1051/m2an/2011075
LA  - en
ID  - M2AN_2012__46_5_1147_0
ER  - 
%0 Journal Article
%A Aurada, Markus
%A Feischl, Michael
%A Praetorius, Dirk
%T Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 1147-1173
%V 46
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an/2011075/
%R 10.1051/m2an/2011075
%G en
%F M2AN_2012__46_5_1147_0
Aurada, Markus; Feischl, Michael; Praetorius, Dirk. Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1147-1173. doi : 10.1051/m2an/2011075. http://archive.numdam.org/articles/10.1051/m2an/2011075/

[1] M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Wiley-Interscience, John Wiley & Sons, New-York (2000). | MR | Zbl

[2] M. Aurada, P. Goldenits and D. Praetorius, Convergence of data perturbed adaptive boundary element methods. ASC Report 40/2009, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2009).

[3] M. Aurada, M. Ebner, M. Feischl, S. Ferraz-Leite, P. Goldenits, M. Karkulik, M. Mayr and D. Praetorius, HILBERT - A Matlab implementation of adaptive 2D-BEM. ASC Report 24/2011, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2011). Software download at http://www.asc.tuwien.ac.at/abem/hilbert/. | Zbl

[4] M. Aurada, S. Ferraz-Leite and D. Praetorius, Estimator reduction and convergence of adaptive BEM. Appl. Numer. Math., in print (2011). | MR | Zbl

[5] I. Babuśka and M. Vogelius, Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44 (1984) 75-102. | MR | Zbl

[6] R. Bank, Hierarchical bases and the finite element method. Acta Numer. 5 (1996) 1-45. | MR | Zbl

[7] F. Bornemann, B. Erdmann and R. Kornhuber, A-posteriori error-estimates for elliptic problems in 2 and 3 space dimensions. SIAM J. Numer. Anal. 33 (1996) 1188-1204. | MR | Zbl

[8] C. Carstensen, An a posteriori error estimate for a first-kind integral equation. Math. Comp. 66 (1997) 139-155. | MR | Zbl

[9] C. Carstensen and D. Praetorius, Averaging techniques for the effective numerical solution of Symm's integral equation of the first kind. SIAM J. Sci. Comput. 27 (2006) 1226-1260. | MR | Zbl

[10] 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) 782-810. | MR | Zbl

[11] C. Carstensen and D. Praetorius, Averaging techniques for a posteriori error control in finite element and boundary element analysis, in Boundary Element Analysis : Mathematical Aspects and Applications, edited by M. Schanz and O. Steinbach. Lect. Notes Appl. Comput. Mech. 29 (2007) 29-59. | MR | Zbl

[12] C. Carstensen and D. Praetorius, Convergence of adaptive boundary element methods. ASC Report 15/2009, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2009). | Zbl

[13] C. Carstensen and E. Stephan, Adaptive coupling of boundary elements and finite elements. ESAIM : M2AN 29 (1995) 779-817. | Numdam | MR | Zbl

[14] M. Costabel, A symmetric method for the coupling of finite elements and boundary elements, in The Mathematics of Finite Elements and Applications IV, MAFELAP 1987, edited by J. Whiteman, Academic Press, London (1988) 281-288. | MR | Zbl

[15] P. Deuflhard, P. Leinen and H. Yserentant, Concepts of an adaptive hierarchical finite element code. Impact Comput. Sci. Eng. 1 (1989) 3-35. | Zbl

[16] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl

[17] W. Dörfler and R. Nochetto, Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002) 1-12. | MR | Zbl

[18] C. Erath, S. Ferraz-Leite, S. Funken and D. Praetorius, Energy norm based a posteriori error estimation for boundary element methods in two dimensions. Appl. Numer. Math. 59 (2009) 2713-2734. | MR | Zbl

[19] C. Erath, S. Funken, P. Goldenits and D. Praetorius, Simple error estimators for the Galerkin BEM for some hypersingular integral equation in 2D. ASC Report 20/2009, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2009). | Zbl

[20] S. Ferraz-Leite and D. Praetorius, Simple a posteriori error estimators for the h-version of the boundary element method. Computing 83 (2008) 135-162. | MR | Zbl

[21] S. Ferraz-Leite, C. Ortner and D. Praetorius, Convergence of simple adaptive Galerkin schemes based on h − h / 2 error estimators. Numer. Math. 116 (2010) 291-316. | MR | Zbl

[22] I. Graham, W. Hackbusch and S. Sauter, Finite elements on degenerate meshes : Inverse-type inequalities and applications. IMA J. Numer. Anal. 25 (2005) 379-407. | MR | Zbl

[23] E. Hairer, S. Nørsett and G. Wanner, Solving ordinary differential equations I, Nonstiff problems. Springer, New York (1987). | MR | Zbl

[24] M. Maischak, P. Mund and E. Stephan, Adaptive multilevel BEM for acoustic scattering. Comput. Methods Appl. Mech. Eng. 150 (2001) 351-367. | MR | Zbl

[25] W. Mclean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR | Zbl

[26] P. Morin, K. Siebert and A. Veeser, A Basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707-737. | MR | Zbl

[27] P. Mund and E. Stephan, An additive two-level method for the coupling of nonlinear FEM-BEM equations. SIAM J. Numer. Anal. 36 (1999) 1001-1021. | MR | Zbl

[28] P. Mund, E. Stephan and J. Weiße, Two-level methods for the single layer potential in R3. Computing 60 (1998) 243-266. | MR | Zbl

[29] S. Rjasanov and O. Steinbach, The fast solution of boundary integral equations. Springer, New York (2007). | MR | Zbl

[30] S. Sauter and C. Schwab, Randelementmethoden : Analyse, Numerik und Implementierung schneller Algorithmen. Teubner Verlag, Wiesbaden (2004). | Zbl

[31] O. Steinbach, Numerical approximation methods for elliptic boundary value problems : Finite and boundary elements. Springer, New York (2008). | MR | Zbl

[32] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner, Stuttgart (1996). | Zbl

[33] E. Zeidler, Nonlinear functional analysis and its applications. part II/B, Springer, New York (1990). | MR | Zbl

Cité par Sources :