Multigrid methods for interface problems

We analyze multigrid convergence when 2D-elliptic boundary value problems with interfaces are discretized using finite element methods where coarse meshes do not approximate the interface geometry. Starting with the initial mesh, the used interface adapted mesh generator constructs a finite element mesh up to a certain refinement level where the interface lines are approximated with a sufficient precision. It is shown that multigrid cycles based on SOR-smoothing and specific interpolation and restriction operators converge independently of the meshsize parameter. Moreover, in practice the convergence is also independent of the ratio of the jumping coefficients. We demonstrate the efficiency of our method by means of numerical examples.