MULTIGRID AND MULTILEVEL METHODS FOR
                      NONCONFORMING ROTATED Q1 ELEMENTS

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

                                 PETER OSWALD
                          Department of Mathematics
                             Texas A&M University
                      College Station, Texas 77843--3368

Abstract.  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