Least Squares Methods for the Coupling of FEM and BEM

Abstract

In the present paper we propose least squares formulations for the numerical solution of exterior boundary value problems. The partial differential equation is a first order system in a bounded subdomain, and the unbounded subdomain is treated by means of boundary integral equations. The first order system is derived from a strongly elliptic second order system. The analysis of the present least squares formulations is reduced to the analysis of the Galerkin method for the coupling of finite element and boundary element methods (FEM and BEM) of the second order problem. The least squares approach requires no stability condition. However, it requires the computation of negative as well as of half integer Sobolev norms. The arising linear systems can be preconditioned to have condition numbers $\sim 1$. The present methods benefit strongly from the use of biorthogonal wavelets on the coupling boundary and the computation of corresponding equivalent norms in Sobolev spaces. In particular, the application of Green's formula leads to an efficient discretization of least squares formulations.