On Multigrid Methods for Generalized Finite Element Methods

This paper reports investigations on how multigrid methods can be applied for the solution of some generalized finite element methods based on the partition of unity technique. One feature of the generalized finite element method is that the underlying algebraic system is often singular due to the overlapping from the partition of unity. While standard iterative methods such as the conjugate gradient method, Jacobi, Gauss-Seidel methods, multigrid methods and domain decomposition methods are still convergent for this type of singular systems, we observe that a standard multigrid method does not converge uniformly with respect to mesh parameters. Using a simple model problem, we will carefully investigate why these method do not work. We will then propose a multigrid method that does converge uniformly as in the standard finite element method.