A least squares coupling method with finite elements and boundary elements for transmission problems (Q1770670)

The authors use standard finite element/boundary element spaces and apply multigrid or the BPX-algorithm [cf. \textit{J. H. Bramble}, \textit{J. E. Pasciak}, and \textit{J. Xu}, Math. Comput. 55, No. 191, 1--22 (1990; Zbl 0703.65076)] to both finite-element and boundary-element discretizations, thus implementing discrete versions of the inner products in \(\tilde H^{-1}(\Omega)\) and \(H^{1/2}(\partial\Omega)\). The model problem is an interface problem for second-order strongly elliptic differential operators in a bounded domain \(\Omega\subset\mathbb{R}^d\) and for the Laplacian in the unbounded domain \(\mathbb R^d\setminus\bar \Omega\) with prescribed jumps \(u_0\) for the displacement and \(t_0\) for its normal derivative on the interface \(\Gamma= \partial\Omega\). The exterior problem is reduced to a strongly elliptic system of integral equations on \(\Gamma\) while the interior problem is reformulated as a first-order system. The flux variable \(\Omega\) is discretized by piecewise constant elements or continuous piecewise linear elements or Raviart-Thomas elements of lower order. The discrete inner products are approximated by the action of multilevel preconditioners and can be used to accelerate the computation of the full discrete least squares system by a preconditioned conjugate gradient method. It is shown that the least squares coupling approach is an efficient and robust solution procedure.