FEM Solution of Exterior Elliptic Problems with Weakly Enforced Integral Non Reflecting Boundary Conditions

Abstract

We consider a coupling of finite element (FEM) and boundary element (BEM) methods for the solution of the Poisson equation in unbounded domains. We propose a numerical method that approximates the solution using computations only in an interior finite domain, bounded by an artificial boundary $${{\mathcal {B}}}$$ . Transmission conditions between the interior domain, discretized by a FEM, and the exterior domain, which is reduced to the boundary $${{\mathcal {B}}}$$ via a BEM, are imposed weakly on $${{\mathcal {B}}}$$ using a mortar approach. The main advantage of this approach is that non matching grids can be used at the interface $${{\mathcal {B}}}$$ of the interior and exterior domains. This allows to exploit the higher accuracy of the BEM with respect to the FEM, which justifies the choice of the discretization in space of the BEM coarser than the one inherited by the spatial discretization of the finite computational domain. We present the analysis of the method and numerical results which show the advantages with respect to the standard approach in terms of computational cost and memory saving.