The Analysis of Intergrid Transfer Operators and Nonconforming Multigrid Methods

Zhangxin Chen

Department of Mathematics, Box 156 Southern Methodist University Dallas, Texas 75275--0156

Abstract

We analyze intergrid transfer operators for nonconforming finite elements and two approaches for defining multigrid methods for discretizations of second and fourth-order elliptic problems using these elements. The first approach is the usual one, which uses discrete equations on all levels which are defined by the same discretization. For this approach we shall find the lower and upper bounds of the energy norm of the coarse-to-fine grid transfer operators for the nonconforming elements considered. The main result of this approach from the analysis of these operators is the convergence of the $\mathcal W$-cycle with any number of smoothing iterations for some of the nonconforming elements, while for others it may diverge unless the number of smoothing iterations on all levels is sufficiently large. The second is based on the Galerkin approach'' where the quadratic forms over coarse grids are constructed from the quadratic form on the finest grid and iterated coarse-to-fine grid operators. To analyze the second approach we shall need to bound the energy norm of these iterated intergrid transfer operators. We shall show the convergence of both the $\mathcal V$-cycle and $\mathcal W$-cycle multigrid methods with any number of smoothing steps when the energy norm of the iterated intergrid operators is bounded. The second approach also applies to partial differential problems without regularity assumptions, while the theory developed here for both approaches carries over directly to mixed finite element methods for partial differential problems as well. Numerical results are presented to illustrate the present theory.