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.
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] and , A posteriori error estimation in finite element analysis. Wiley-Interscience, John Wiley & Sons, New-York (2000). | MR | Zbl
[2] , and , 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] , , , , , , and , 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] , and , Estimator reduction and convergence of adaptive BEM. Appl. Numer. Math., in print (2011). | MR | Zbl
[5] and , Feedback and adaptive finite element solution of one-dimensional boundary value problems. Numer. Math. 44 (1984) 75-102. | MR | Zbl
[6] , Hierarchical bases and the finite element method. Acta Numer. 5 (1996) 1-45. | MR | Zbl
[7] , and , A-posteriori error-estimates for elliptic problems in 2 and 3 space dimensions. SIAM J. Numer. Anal. 33 (1996) 1188-1204. | MR | Zbl
[8] , An a posteriori error estimate for a first-kind integral equation. Math. Comp. 66 (1997) 139-155. | MR | Zbl
[9] and , 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] and , 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] and , 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] and , Convergence of adaptive boundary element methods. ASC Report 15/2009, Institute for Analysis and Scientific Computing, Vienna University of Technology, Wien (2009). | Zbl
[13] and , Adaptive coupling of boundary elements and finite elements. ESAIM : M2AN 29 (1995) 779-817. | Numdam | MR | Zbl
[14] , 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] , and , Concepts of an adaptive hierarchical finite element code. Impact Comput. Sci. Eng. 1 (1989) 3-35. | Zbl
[16] , A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33 (1996) 1106-1124. | MR | Zbl
[17] and , Small data oscillation implies the saturation assumption. Numer. Math. 91 (2002) 1-12. | MR | Zbl
[18] , , and , Energy norm based a posteriori error estimation for boundary element methods in two dimensions. Appl. Numer. Math. 59 (2009) 2713-2734. | MR | Zbl
[19] , , and , 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] and , Simple a posteriori error estimators for the h-version of the boundary element method. Computing 83 (2008) 135-162. | MR | Zbl
[21] , and , Convergence of simple adaptive Galerkin schemes based on h − h / 2 error estimators. Numer. Math. 116 (2010) 291-316. | MR | Zbl
[22] , and , Finite elements on degenerate meshes : Inverse-type inequalities and applications. IMA J. Numer. Anal. 25 (2005) 379-407. | MR | Zbl
[23] , and , Solving ordinary differential equations I, Nonstiff problems. Springer, New York (1987). | MR | Zbl
[24] , and , Adaptive multilevel BEM for acoustic scattering. Comput. Methods Appl. Mech. Eng. 150 (2001) 351-367. | MR | Zbl
[25] , Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000). | MR | Zbl
[26] , and , A Basic convergence result for conforming adaptive finite elements. Math. Models Methods Appl. Sci. 18 (2008) 707-737. | MR | Zbl
[27] and , An additive two-level method for the coupling of nonlinear FEM-BEM equations. SIAM J. Numer. Anal. 36 (1999) 1001-1021. | MR | Zbl
[28] , and , Two-level methods for the single layer potential in R3. Computing 60 (1998) 243-266. | MR | Zbl
[29] and , The fast solution of boundary integral equations. Springer, New York (2007). | MR | Zbl
[30] and , Randelementmethoden : Analyse, Numerik und Implementierung schneller Algorithmen. Teubner Verlag, Wiesbaden (2004). | Zbl
[31] , Numerical approximation methods for elliptic boundary value problems : Finite and boundary elements. Springer, New York (2008). | MR | Zbl
[32] , A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner, Stuttgart (1996). | Zbl
[33] , Nonlinear functional analysis and its applications. part II/B, Springer, New York (1990). | MR | Zbl
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