Convergence of some adaptive FEM-BEM coupling for elliptic but possibly nonlinear interface problems
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 5, p. 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 : https://doi.org/10.1051/m2an/2011075
Classification:  65N30,  65N15,  65N38
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {5},
     year = {2012},
     pages = {1147-1173},
     doi = {10.1051/m2an/2011075},
     zbl = {1276.65066},
     mrnumber = {2916376},
     language = {en},
     url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 5, pp. 1147-1173. doi : 10.1051/m2an/2011075. http://www.numdam.org/item/M2AN_2012__46_5_1147_0/

[1] M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Wiley-Interscience, John Wiley & Sons, New-York (2000). | MR 1885308 | Zbl 1008.65076

[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 1298.65183

[4] M. Aurada, S. Ferraz-Leite and D. Praetorius, Estimator reduction and convergence of adaptive BEM. Appl. Numer. Math., in print (2011). | MR 2908795 | Zbl 1237.65131

[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 745088 | Zbl 0574.65098

[6] R. Bank, Hierarchical bases and the finite element method. Acta Numer. 5 (1996) 1-45. | MR 1624587 | Zbl 0865.65078

[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 1393909 | Zbl 0863.65069

[8] C. Carstensen, An a posteriori error estimate for a first-kind integral equation. Math. Comp. 66 (1997) 139-155. | MR 1372001 | Zbl 0854.65102

[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 2199747 | Zbl 1105.65124

[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 2306268 | Zbl 1140.65089

[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 2298798 | Zbl 1298.65161

[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 1238.65124

[13] C. Carstensen and E. Stephan, Adaptive coupling of boundary elements and finite elements. ESAIM : M2AN 29 (1995) 779-817. | Numdam | MR 1364401 | Zbl 0849.65083

[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 956899 | Zbl 0682.65069

[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 0706.65111

[16] W. Dörfler, A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR 1393904 | Zbl 0854.65090

[17] W. Dörfler and R. Nochetto, Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002) 1-12. | MR 1896084 | Zbl 0995.65109

[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 2566766 | Zbl 1177.65192

[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 1278.65168

[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 2529605 | Zbl 1175.65126

[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 2672266 | Zbl 1198.65213

[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 2126208 | Zbl 1076.65098

[23] E. Hairer, S. Nørsett and G. Wanner, Solving ordinary differential equations I, Nonstiff problems. Springer, New York (1987). | MR 868663 | Zbl 0789.65048

[24] M. Maischak, P. Mund and E. Stephan, Adaptive multilevel BEM for acoustic scattering. Comput. Methods Appl. Mech. Eng. 150 (2001) 351-367. | MR 1487950 | Zbl 0916.76042

[25] W. Mclean, Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR 1742312 | Zbl 0948.35001

[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 2413035 | Zbl 1153.65111

[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 1694625 | Zbl 0938.65138

[28] P. Mund, E. Stephan and J. Weiße, Two-level methods for the single layer potential in R3. Computing 60 (1998) 243-266. | MR 1621301 | Zbl 0901.65072

[29] S. Rjasanov and O. Steinbach, The fast solution of boundary integral equations. Springer, New York (2007). | MR 2310663 | Zbl 1119.65119

[30] S. Sauter and C. Schwab, Randelementmethoden : Analyse, Numerik und Implementierung schneller Algorithmen. Teubner Verlag, Wiesbaden (2004). | Zbl 1059.65108

[31] O. Steinbach, Numerical approximation methods for elliptic boundary value problems : Finite and boundary elements. Springer, New York (2008). | MR 2361676 | Zbl 1153.65302

[32] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner, Stuttgart (1996). | Zbl 0853.65108

[33] E. Zeidler, Nonlinear functional analysis and its applications. part II/B, Springer, New York (1990). | MR 1033498 | Zbl 0684.47029