This paper is concerned with the dual formulation of the interface problem consisting of a linear partial differential equation with variable coefficients in some bounded Lipschitz domain Ω in (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := . The two problems are coupled by transmission and Signorini contact conditions on the interface Γ = ∂Ω. The exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. The resulting variational formulation becomes a variational inequality with a linear operator. Then we treat the corresponding numerical scheme and discuss an approximation of the NtD mapping with an appropriate discretization of the inverse Poincaré-Steklov operator. In particular, assuming some abstract approximation properties and a discrete inf-sup condition, we show unique solvability of the discrete scheme and obtain the corresponding a-priori error estimate. Next, we prove that these assumptions are satisfied with Raviart-Thomas elements and piecewise constants in Ω, and continuous piecewise linear functions on Γ. We suggest a solver based on a modified Uzawa algorithm and show convergence. Finally we present some numerical results illustrating our theory.
Mots-clés : Raviart-Thomas space, boundary integral operator, Lagrange multiplier
@article{M2AN_2011__45_4_779_0,
author = {Gatica, Gabriel N. and Maischak, Matthias and Stephan, Ernst P.},
title = {Numerical analysis of a transmission problem with {Signorini} contact using {mixed-FEM} and {BEM}},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {779--802},
publisher = {EDP-Sciences},
volume = {45},
number = {4},
year = {2011},
doi = {10.1051/m2an/2010102},
mrnumber = {2804659},
zbl = {1267.74110},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2010102/}
}
TY - JOUR AU - Gatica, Gabriel N. AU - Maischak, Matthias AU - Stephan, Ernst P. TI - Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2011 SP - 779 EP - 802 VL - 45 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2010102/ DO - 10.1051/m2an/2010102 LA - en ID - M2AN_2011__45_4_779_0 ER -
%0 Journal Article %A Gatica, Gabriel N. %A Maischak, Matthias %A Stephan, Ernst P. %T Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2011 %P 779-802 %V 45 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2010102/ %R 10.1051/m2an/2010102 %G en %F M2AN_2011__45_4_779_0
Gatica, Gabriel N.; Maischak, Matthias; Stephan, Ernst P. Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 45 (2011) no. 4, pp. 779-802. doi : 10.1051/m2an/2010102. http://www.numdam.org/articles/10.1051/m2an/2010102/
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