Efficient numerical methods for quad-curl problem and fast solvers

Abstract

The quad-curl problem is an important mathematical model abstract from the magnetohydrodynamics problem and Maxwell's transmission eigenvalue problem. This thesis is concerned with efficient numerical methods for the quad-curl problem and fast solvers. First, the C0 interior penalty method is investigated for the quad-curl problem model. Then, efficient multigrid algorithms are proposed for the quad-curl problem based on the C0 interior penalty method. In addition, a new mixed finite element method is studied for obtaining numerical approximated solutions of the quad-curl problem. We begin this thesis with two model problems that lead to quad-curl problem as well as a brief survey of finite element method for this problem. As a comparable example, we review some results for the biharmonic problem, including regularity results, finite element methods and multigrid algorithms based on the C0 interior penalty method. The applications of these methods provide not only an innovation in methods but also a possible way to solve the new quad-curl problem numerically. The thesis consists of three main parts. The first part aims to study a C0 interior penalty method for the quad-curl problem with relatively a simple discrete structure. It also produces optimal error estimates in the corresponding discrete energy norm. This numerical scheme is proved to be well proposed. Meanwhile, a C1 approximate solution is obtained by using a post-processing procedure. In the second part, we investigate the multigrid methods for the quad-curl problem, including W -cycle and V -cycle algorithms, based on the C0 interior penalty method. We produce an efficient preconditioner for each smoothing step and prove that both the W -cycle and V -cycle algorithms can converge in an effective way, theoretically and numerically. In the third part of the thesis, a new mixed finite element method with a perturbed term is developed for the quad-curl problem. Two finite element spaces based on edge elements are constructed so that the discrete mixed structure is proved to be well-posed on them. An effective error analysis strategy for the new mixed finite element scheme is derived such that optimal error estimates are obtained. Numerical experiments are performed to verity all theoretical results.