Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 4, p. 779-802

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 ${ℝ}^{n}$ (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := ${ℝ}^{n}\setminus \overline{\Omega }$. 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.

DOI : https://doi.org/10.1051/m2an/2010102
Classification:  65N30,  65N38,  65N22,  65F10
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {45},
number = {4},
year = {2011},
pages = {779-802},
doi = {10.1051/m2an/2010102},
zbl = {1267.74110},
mrnumber = {2804659},
language = {en},
url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 45 (2011) no. 4, pp. 779-802. doi : 10.1051/m2an/2010102. http://www.numdam.org/item/M2AN_2011__45_4_779_0/

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