Convergence of multigrid methods for nonconforming finite elements without regularity assumptions

Abstract

In this paper Ne consider multigrid methods for discretizations of second-order elliptic problems using nonconforming finite elements. These multigrid methods are based on the "Galerkin approach" where the quadratic forms over coarse grids are constructed from the quadratic form on the finest grid and the iterated coarse-to-fine grid operators. In terms of this approach we show that the nonconforming multigrid operator can be written in a product form, which, together with an appropriate approximation hypothesis, implies convergence estimates for both the V-cycle and W-cycle multigrid methods for partial differential problems without regularity assumptions. These estimates can thus be applied to problems with rough coefficients and local grid refinement. Numerical results are also presented to illustrate the present theory.