On the convergence of Galerkin-multigrid methods for nonconforming finite elements

In this paper we analyze a class of multigrid methods for discretizations of partial differential problems using nonconforming finite elements. We define these multigrid methods in terms of the `Galerkin approach' where quadratic forms over coarse grids are constructed from the quadratic form on the finest grid and iterated coarse-to-fine intergrid transfer operators, which we call Galerkin-multigrid methods. With this approach, we show that uniform convergence rates (independently of the number of levels) can be obtained for both V and W-cycle multigrid methods with one smoothing per level for the nonconforming P1 and rotated Q1 finite elements for second-order problems. We also discuss extensions of this approach to other nonconforming elements such as the Morley, Zienkiewicz, and Adini elements for fourth-order problems. The present methods apply to mixed finite elements for the problems under consideration. Numerical results are presented to compare various approaches for defining multigrid methods for nonconforming finite elements and to show the behavior of the energy norm of the iterates of intergrid transfer operators for these elements. Differential problems with less than full elliptic regularity and without regularity are considered.