A BEM-FEM overlapping algorithm for the Stokes equation

In this work we study the numerical solution of the Dirichlet problem for the Stokes equation by overlapping boundary and finite elements. The domain where we state our problem is the intersection of a polyhedron with the exterior of a strictly contained obstacle with smooth boundary. The solution is then decoupled as a sum of an incident flow, defined on the polyhedron domain plus the response of the interior obstacle expressed as a single layer potential and constructed only on the exterior of the obstacle. The numerical algorithm follows closely this ansatz by replacing the continuous terms of this decomposition by a finite and a boundary element respectively. We prove that, under not very restrictive assumptions, the method is well defined and converges to the exact solution with the same order as the best approximation of the solution by the discrete spaces in the natural norms of the problem. Finally, in the last section we show how this method can be implemented, overcoming some of the difficulties appearing in the implementation, and demonstrating its applicability to practical problems.