# 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 (2011)

- 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 45.4 (2011): 779-802. <http://eudml.org/doc/197474>.

@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 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.
},

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

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

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

language = {eng},

month = {2},

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/197474},

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

DA - 2011/2//

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 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.

LA - eng

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

UR - http://eudml.org/doc/197474

ER -

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