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.

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 (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc := n Ω ¯. 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 : 10.1051/m2an/2010102
Classification : 65N30, 65N38, 65N22, 65F10
Mots clés : Raviart-Thomas space, boundary integral operator, Lagrange multiplier
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     title = {Numerical analysis of a transmission problem with {Signorini} contact using {mixed-FEM} and {BEM}},
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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://archive.numdam.org/articles/10.1051/m2an/2010102/

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