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Numerical analysis of a transmission problem with Signorini contactusing mixed-FEM and BEM*
Published online by Cambridge University Press: 21 February 2011
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.
- Type
- Research Article
- Information
- ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 4 , July 2011 , pp. 779 - 802
- Copyright
- © EDP Sciences, SMAI, 2011
References
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