University of Nijmegen, The Netherlands

Based on a study of convergence proofs for nonconforming multigrid methods, we
introduce a new, more effective smoother and prolongation for the Morley
discretization of the biharmonic equation, or equivalently, for the
(P_{0}, nonconforming P_{1}) discretization of the Stokes
problem. In our experiments, even the V-cycle with one pre- and
post-smoothing turns out to yield uniformly bounded (and small) condition
numbers, whereas the costs are equal to such a cycle with standard
prolongation and Richardson smoother.

The relation between above discretizations of biharmonic and Stokes problems
has been used more often to develop and analyze multi-level methods, in the
sense that the analysis took place in the framework of the biharmonic problem.
We will follow the opposite approach. Since in contrast to the biharmonic
operator, the Stokes operator has full regularity, suitable smoothers
developed in the latter framework can be expected to be more effective. In
particular, an asymptotic bound ~nu^{-1} for the contraction number of
the W-cycle can be shown, where nu is the number of smoothing steps, whereas
in the framework of the biharmonic equation, a bound ~nu^{-1/2} is the
best one can hope for. On the other hand, since in the Stokes framework the
standard basis is not L^{2}-stable, to show the smoothing property we
had to develop a non-standard smoother involving a conforming multigrid call
on a subset of the unknowns.

The standard prolongation used for the Morley finite element space has energy norm larger than 2. This means that the corresponding coarse-grid correction is divergent, and as a consequence, combined with a relatively poor smoother it may even result in a divergent W-cycle. The analysis of additive nonconforming multi-level methods has shown that the energy norms of the iterated prolongations are relevant; (sub-)optimal results can only be expected when these norms are uniformly bounded. However, again for the standard Morley prolongation these norms grow exponentially. Therefore, we introduce a new prolongation based on some local energy minimalization, which at the same time reproduces second order polynomials. In our experiments, the energy norm of this prolongation is less than 2, and the energy norms of the iterated prolongations are uniformly bounded. In our practical algorithm we do not implement this prolongation, but instead we let the prolongation follow by a block Gauss-Seidel step on the sets where we want the energy to be minimized. This results both in a good virtually prolongation and in an effective smoother.