Multigrid Methods for a Class of Saddle Point Problems

点击翻译按钮获取中文摘要

The fast multigrid methods developed and studied in this work are expected to have a broader impact on the numerical solutions of a large class of practical problems. Important applications include: vector (Hodge) Laplacian, Maxwell equations, Stokes equations, Oseen and Navier-Stokes equations, and Magnetohydrodynamics (MHD) etc. MHD, in particular, has important applications in the development of fusion technology and casting processes. In these applications, since no experimentation is nowadays possible, the numerical simulation of the corresponding partial differential equations is indispensable. These simulations are very challenging, requiring large computational resources. The multigrid solvers developed in this project offer the potential for increasingly accurate models to be solved. In addition, our improvements in algorithm developments will have impact on many other areas, such as image processing, and computer graphics.This project is divided into two parts: algorithmic development and theoretical analysis. For the algorithmic development, multigrid solvers will be developed for mixed finite element discretization based on Finite Element Exterior Calculus (FEEC). In our study, effective smoothers, which are the key of multigrid methods, will be developed by using existing effective preconditioners or splitting schemes. One such example is a distributive smoother proposed in this project which is highly related to the well-known projection methods used in computational fluid dynamic. In addition to the algorithmic development, a more completed convergence theory of multigrid methods for saddle point problems will be developed. This theory aims to relax the strong regularity assumption in existing work. Consequently our theory can be applied to more realistic problems especially for solutions with singularities. Our theoretical investigation will also provide insight for the algorithmic development, e.g., the construction of approximated distributive smoothers and Schwarz smoothers, and optimal choice of relaxation parameters used in several smoothers.