Analysis of finite element approximation and iterative methods for time-dependent Maxwell problems

Abstract

In this dissertation we are concerned with the analysis of the finite element method for the time-dependent Maxwell interface problem when Nedelec and Raviart-Thomas finite elements are employed and preconditioning of the resulting linear system when implicit time schemes are used.

We first investigate the finite element method proposed by Makridakis and Monk in 1995. After studying the regularity of the solution to time dependent Maxwell's problem and providing approximation estimates for the Fortin operator, we are able to give the optimal error estimate for the semi-discrete scheme for Maxwell's equations.

Then we study preconditioners for linear systems arising in the finite element method for time-dependent Maxwell's equations using implicit time-stepping. Such linear systems are usually very large but sparse and can only be solved iteratively. We consider overlapping Schwarz methods and multigrid methods and extend some existing theoretical convergence results. For overlapping Schwarz methods, we provide numerical experiments to confirm the theoretical analysis.