New Non-classical Finite Element Methods for Maxwell Equations

Dr. Fengyan Li

Department of Mathematics
University of South Carolina

February 17, 2006
202 Madison Hall
1:00 PM (Coffee and Cookies at 12:30)

Abstract

Electromagnetism has been an indispensable part of many current technologies such as wireless communication, medical scan machine, radar, high-frequency/ high-speed circuits, among many others. Since 1873 when Maxwell founded the modern theory of electromagnetism by formulating the equations that now bear his name, electromagnetism becomes an active research field in many engineering and scientific studies. The problems in this area quite often are characterized by the geometry complexity, large size in terms of characteristic wavelength, heterogeneous materials, divergence constraints satisfied by the fields and the existence of singularities related to the geometric boundaries or material interfaces. Therefore robust high-order methods with flexible requirement on meshes and local approximations are favored in order to model the wave phenomena over long time, or in the geometrically complicated domains. On the other hand, reducing the continuity requirements on the numerical solutions across the element interfaces is one approach which may lead to useful approximation solutions around the geometric corners. Due to these concerns, we develop two types of non-classical finite element methods. They are the locally divergence-free discontinuous Galerkin methods for time-dependent Maxwell equations and the locally divergence-free non-conforming finite element method for reduced time-harmonic Maxwell equations. I will explain these methods in this talk and present the main theoretical results along with the numerical examples which demonstrate the performance of these methods.