In this paper we systematically study multigrid algorithms and multilevel preconditioners for discretizations of second-order elliptic problems using nonconforming rotated Q1 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