Department of Mathematics

Box 156

Southern Methodist University

Dallas, Texas 75275--0156

Department of Mathematics

Texas A&M University

College Station, TX 77843

In this paper we systematically study multigrid algorithms and multilevel
preconditioners for discretizations of second-order elliptic problems using
nonconforming rotated Q_{1} finite elements. We first derive optimal
results for the W-cycle and variable V-cycle multigrid algorithms; we prove
that the W-cycle algorithm with a sufficiently large number of smoothing steps
converges in the energy norm at a rate which is independent of grid number
levels, and that the variable V-cycle algorithm provides a preconditioner with
a condition number which is bounded independently of the number of grid
levels. In the case of constant coefficients, the optimal convergence
property of the W-cycle algorithm is shown with any number of smoothing steps.
Then we obtain suboptimal results for multilevel additive and multiplicative
Schwarz methods and their related V-cycle multigrid algorithms; we show that
these methods generate preconditioners with a condition number which can be
bounded at least by the number of grid levels. Also, we consider the problem
of switching the present discretizations to spectrally equivalent
discretizations for which optimal preconditioners already exist. Finally, the
numerical experiments carried out here complement these theories.

This paper is in the series of ISC-95-10-Math Technical Reports, Texas A&M University. It is available on http://www.isc.tamu.edu

Contributed March 2, 1996.