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.