An analysis of nonconforming multi-grid methods, leading to an improved method for the Morley element

Authors:
Rob Stevenson
Affiliations:
Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands
Venue:
Mathematics of Computation
Year:
2003

Citing 7
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Abstract

We recall and slightly refine the convergence theory for nonconforming multi-grid methods for symmetric positive definite problems developed by Bramble, Pasciak and Xu. We derive new results to verify the regularity and approximation assumption, and the assumption on the smoother. From the analysis it will appear that most efficient multi-grid methods can be expected for fully regular problems, and for prolongations for which the energy norm of the iterated prolongations is uniformly bounded.Guided by these observations, we develop a new multi-grid method for the biharmonic equation discretized with Morley finite elements, or equivalently, for the Stokes equations discretized with the P0-nonconforming P1 pair. Numerical results show that the new method is superior to standard ones.