A Least Squares Coupling Method with Finite Elements and Boundary Elements for Transmission Problems

摘要

We analyze a least squares formulation for the numerical solution of second-order linear transmission problems in two and three dimensions, which allow jumps on the interface. In a bounded domain the second-order partial differential equation is rewritten as a first-order system; the part of the transmission problem which corresponds to the unbounded exterior domain is reformulated by means of boundary integral equations on the interface. The least squares functional is given in terms of Sobolev norms of order -1 and of order 1/2. These norms are computed by approximating the corresponding inner products using multilevel preconditioners for a second-order elliptic problem in a bounded domain Ω and for the weakly singular integral operator of the single layer potential on its boundary αΩ. As preconditioners we use both multigrid and BPX algorithms, and the preconditioned system has bounded or mildly growing condition number Numerical experiments confirm our theoretical results.