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

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

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

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

Abstract

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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 ∖ 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 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.

How to cite

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Gatica, 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 -

References

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