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

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

- Volume: 45, Issue: 4, page 779-802
- ISSN: 0764-583X

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topGatica, Gabriel N., Maischak, Matthias, and 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 45.4 (2011): 779-802. <http://eudml.org/doc/273192>.

@article{Gatica2011,

abstract = {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 $\mathbb \{R\}^n$ (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc:= $\mathbb \{R\}^n\backslash \bar\{\Omega \}$. 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 correspondinga-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.},

author = {Gatica, Gabriel N., Maischak, Matthias, Stephan, Ernst P.},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Raviart-Thomas space; boundary integral operator; Lagrange multiplier},

language = {eng},

number = {4},

pages = {779-802},

publisher = {EDP-Sciences},

title = {Numerical analysis of a transmission problem with Signorini contact using mixed-FEM and BEM},

url = {http://eudml.org/doc/273192},

volume = {45},

year = {2011},

}

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 - Modélisation Mathématique et Analyse Numérique

PY - 2011

PB - EDP-Sciences

VL - 45

IS - 4

SP - 779

EP - 802

AB - 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 $\mathbb {R}^n$ (n ≥ 2) and the Laplace equation with some radiation condition in the unbounded exterior domain Ωc:= $\mathbb {R}^n\backslash \bar{\Omega }$. 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 correspondinga-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.

LA - eng

KW - Raviart-Thomas space; boundary integral operator; Lagrange multiplier

UR - http://eudml.org/doc/273192

ER -

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