Multigrid Algorithms For Nonconforming And Mixed Methods For Symmetric And Nonsymmetric Problems

. In this paper we consider multigrid algorithms for nonconforming and mixed finite element methods for symmetric and nonsymmetric second order elliptic problems. We prove optimal convergence properties of the W-cycle multigrid algorithm and uniform condition number estimates for the variable V-cycle preconditioner for the symmetric problem. For the nonsymmetric and/or indefinite problem, we show that a simple V-cycle multigrid iteration converges at a uniform rate provided that the coarsest level in the multilevel iteration is sufficiently fine (but independent on the number of multigrid levels). Various types of smoothers for the nonsymmetric and indefinite problem are discussed. Extensive numerical results for both symmetric and nonsymmetric problems are given to illustrate the present theories. Key words. mixed method, nonconforming method, finite elements, multigrid algorithm, convergence, symmetric problems, nonsymmetric and/or indefinite problems AMS(MOS) subject classificatio..