Multigrid Methods for Nonconforming Finite Elements

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8904911 Brenner The multigrid method for finite elements is an optimal order algorithm for solving elliptic boundary value problems. The error of the approximate solution lies within the theoretical bound for the global error, yet the cost of computation is proportional only to the number of unknowns in the discretized equations. Until recently, most of the research in this area concentrated primarily on nested conforming finite element spaces. However, nonconforming finite elements are often used in applications because they are easier to program. In previous work, results on full W-cycle multigrid convergence for nonconforming finite elememts have been obtained by the principal investigator in the study of P1 nonconforming finite element applied to the Poisson equation and the Morley finite element applied to the biharmonic equation on polygonal domains. In this project, these results will be applied to the stationary Stokes equations and extended to domains with smooth boundary. Multigrid algorithms will be developed for nonconforming methods in linear elasticity and plate bending and also for mixed methods that are equivalent to nonconforming methods. The convergence of V-cycle nonconforming methods will be investigated. Numerical experiments will be performed for each of these problems.