Numerical analysis of a non-linear transmission problem with Signorini contact using dual-dual mixed-FEM and BEM — a priori and a posteriori error estimates

In this paper we generalize the approach in [5] and discuss an interface problem consisting of a non-linear partial differential equation in Ω ⊂ Rn (bounded, Lipschitz, n ≥ 2) and the Laplace equation in the unbounded exterior domain Ωc := R n\Ω̄ fulfilling some radiation condition, which are coupled by transmission conditions and Signorini conditions imposed on the interface. The interior pde is discretized by a mixed formulation, whereas the exterior part of the interface problem is rewritten using a Neumann to Dirichlet mapping (NtD) given in terms of boundary integral operators. We treat the general numerical approximation of the resulting variational inequality and discuss the non-trivial discretization of the NtD mapping. Assuming some abstract approximation properties and a discrete inf-sup condition we prove existence and uniqueness and show an a-priori estimate, which generalizes the results in [5]. Choosing Raviart-Thomas elements and piecewise constants in Ω and hat functions on ∂Ω the discrete inf-sup condition is satisfied [1, 3]. We present a solver based on a modified Uzawa algorithm, reducing the solution procedure of the non-linear saddle point problem with an inequality constraint to the repeated solution of a standard non-linear saddle point problem and the solution of a variational inequality based on an elliptic operator. Finally, we present a residual based a-posteriori error estimator compatible with the Signorini condition and a corresponding adaptive scheme, see [6]. Some numerical experiments are shown which illustrate the convergence behavior of the uniform h-version with triangles and rectangles and the adaptive scheme as well as the bounded iteration numbers of the modified Uzawa algorithm, underlining the theoretical results.