Nonconforming multigrid method for nonsymmetric and indefinite problems

Abstract

This thesis is devoted to numerical analysis of multigrid methods for solving linear systems arising from the approximation to partial differential equations using P1-nonconforming finite elements. There have been two types of nonconforming multigrid algorithms for solving the symmetric positive definite problems. The first one exploits the nonconforming finite elements in both smoothing iterations and coarse-grid corrections in the multilevel iteration. The second one uses the nonconforming finite elements in the smoothing iterations on the finest level, but conforming finite elements in the coarse-grid corrections. We extend the nonconforming multigrid methods to the nonsymmetric and/or indefinite problems and show that the rate of convergence is optimal. We also consider the two-level additive Schwarz preconditioner for nonconforming finite element spaces applied to nonsymmetric and/or indefinite problems and show that the rate of convergence is optimal. We also consider the two-level additive Schwarz preconditioner for nonconforming finite element spaces applied to nonsymmetric and/or indefinite problems. We show that the rate of convergence is independent of the number of degrees of freedom and the number of local problems if the coarse mesh is fine enough.