An Efficient Multigrid Method for the Morley Discretization of the Biharmonic Equation

Rob Stevenson
University of Nijmegen, The Netherlands

Abstract

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 (P0, nonconforming P1) 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 L2-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.