The Analysis of Intergrid Transfer Operators and Nonconforming Multigrid Methods

In this paper we rst analyze intergrid transfer operators and their iterates for some nonconforming nite elements used for discretizations of second and fourth-order elliptic problems. Then two classes of multigrid methods using these elements are considered. The rst class is the usual one, which uses discrete equations on all levels which are deened by the same discretiza-tion, while the second one is based on the Galerkin approach where quadratic forms over coarse grids are constructed from the quadratic form on the nest grid and the iterates of intergrid transfer operators, which we call the Galerkin multigrid method. The properties of these intergrid transfer operators are utilized for the analysis of the rst class, while the properties of their iterates are exploited for the second one. Convergence results available for these two classes of multigrid methods are summarized here.