Overcoming corner singularities using multigrid methods

We consider the Poisson equation -Delta u = f with homogeneous Dirichlet boundary condition on a two-dimensional polygonal domain Omega. We develop a finite element multigrid method on quasi-uniform grids that obtains O(h(m+1?epsilon)) convergence in the H-1 (Omega) norm for any positive O epsilon when f is an element of H-m (Omega). The cost of the method is proportional to the number of elements in the triangulation. The results of this paper can be generalized to other equations and other boundary conditions.