Multigrid And DDM For Mortar-type Nonconforming Quadrilateral Elements

In this thesis, we utilize the mortar-type nonconforming quadrilateral elements for the discretization of elliptic problems. In the Chapter 1, we consider multigrid methods for the mortar-type nonconforming quadrilateral element proposed in [20]. An intergrid transfer operator is proposed. Based on the theory developed by [6], it is proven that the W-cycle multigrid method is optimal, i.e., the convergence rate is independent of the mesh size and mesh level. Meanwhile, it is proven that the condition number of the variable V-cycle multigrid preconditioner is uniformly bounded. Numerical experiments demonstrate that the optimal convergence property for the W-cycle algorithm holds with any number of smoothing steps. In the Chapter 2, we consider domain decomposition methods for quadrilateral elements. We provide additive Schwarz methods for the mortar-type rotated Q₁ element, and prove that the condition number of the additive Schwarz preconditioned operator is proportional to (1 + log(H/h)), where H is the coarse mesh size and h is the fine mesh size.