Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM*
CIMA and Departamento de Ingeniería
Matemática, Universidad de Concepción, Casilla 160-C,
Concepción, Chile. email@example.com
2 BICOM, Brunel University, UB8 3PH, Uxbridge, UK. firstname.lastname@example.org
3 Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167 Hannover, Germany. email@example.com
Revised: 28 October 2010
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.
Mathematics Subject Classification: 65N30 / 65N38 / 65N22 / 65F10
Key words: Raviart-Thomas space / boundary integral operator / Lagrange multiplier
This research was partially supported by FONDAP and BASAL projects CMM, Universidad de Chile, by Centro de Investigación en Ingeniería Matemática (CI2MA) of the Universidad de Concepción, by the German Academic Exchange Service (DAAD) through the project 412/HP-hys-rsch, and by the German Research Foundation (DFG) under grant Ste 573/3.
© EDP Sciences, SMAI, 2011