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

Author: Rob Stevenson
Journal: Math. Comp. 72 (2003), 55-81
MSC (2000): Primary 65N55, 65N30, 65F10
Published electronically: May 1, 2002
MathSciNet review: 1933814
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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 $P_0$-nonconforming $P_1$pair. Numerical results show that the new method is superior to standard ones.

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Additional Information

Rob Stevenson
Affiliation: Department of Mathematics, Utrecht University, P.O. Box 80.010, NL-3508 TA Utrecht, The Netherlands

Keywords: Multi-grid method, nonconforming finite elements, biharmonic equation, Morley finite element space, Stokes equations
Received by editor(s): November 23, 1998
Received by editor(s) in revised form: January 23, 2001
Published electronically: May 1, 2002
Article copyright: © Copyright 2002 American Mathematical Society