Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM

Gatica, Gabriel N.; Maischak, Matthias; Stephan, Ernst P.

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

Classification : 65N30, 65N38, 65N22, 65F10
Mots clés : Raviart-Thomas space, boundary integral operator, Lagrange multiplier

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