Mathematics Department

Cornell University

Ithaca, NY

A multigrid preconditioning scheme for solving the Ciarlet-Raviart mixed
method equations for the biharmonic Dirichlet problem is presented. In
particular, a Schur complement formulation for these equations which yields
non-inherited quadratic forms is considered. The preconditioning scheme is
compared with a standard W-cycle multigrid iteration. It is proved that a
Variable V-cycle preconditioner leads to problems with uniformly bounded
condition numbers. However, W-cycle convergence is proved only if the number
of smoothings "m is sufficiently large". An example is given in which the
W-cycle diverges unless "m > 7". Divergent W-cycles are also encountered when
solving the Morley equations for the biharmonic Dirichlet problem; although,
Brenner has proved W-cycle convergence for sufficiently large "m"
[Brenner,(1989)]. This is illustrated with additional computations, while
Variable V-cycles continue to produce excellent preconditioners in this
setting.
Certain approximate L_{2}-inner products are described and a
modification to the Ciarlet-Raviart method is proposed which reduces the work
of the multilevel schemes. Optimal order error estimates are proved for the
modified method. Consideration is restricted to Ciarlet-Raviart methods of
quadratic and higher degree throughout the paper.

Contributed March 25, 1992, updated April 17, 1992.