Application of Helmholtz/Hodge Decomposition to Finite Element Methods for Two-Dimensional Maxwell's Equations

In this work we apply the two-dimensional Helmholtz/Hodge decomposition to develop new finite element schemes for two-dimensional Maxwell's equations. We begin with the introduction of Maxwell's equations and a brief survey of finite element methods for Maxwell's equations. Then we review the related fundamentals in Chapter 2. In Chapter 3, we discuss the related vector function spaces and the Helmholtz/Hodge decomposition which are used in Chapter 4 and 5. The new results in this dissertation are presented in Chapter 4 and Chapter 5. In Chapter 4, we propose a new numerical approach for two-dimensional Maxwell's equations that is based on the Helmholtz/Hodge decomposition for divergence-free vector fields. In this approach an approximate solution for Maxwell's equations can be obtained by solving standard second order scalar elliptic boundary value problems. This new approach is illustrated by a P1 finite element method. In Chapter 5, we further extend the new approach described in Chapter 4 to the interface problem for Maxwell's equations. We use the extraction formulas and multigrid method to overcome the low regularity of the solution for the Maxwell interface problem. The theoretical results obtained in this dissertation are confirmed by numerical experiments.

Name thesis:degree_name

Doctor of Philosophy (PhD)

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Applied Mathematics

Grantor

Mathematics

Year dc:date.available

2013

Author dc:creator

• Nan, Zhe

• multigrid algorithms
• Helmholtz/Hodge decomposition
• Maxwell's equations
• interface problems
• extraction formulas
• finite element methods

Statement dc:rights

• withheld
• Secure the entire work for patent and/or proprietary purposes for a period of one year. Student has submitted appropriate documentation which states: During this period the copyright owner also agrees not to exercise her/his ownership rights, including public use in works, without prior authorization from LSU. At the end of the one year period, either we or LSU may request an automatic extension for one additional year. At the end of the one year secure period (or its extension, if such is requested), the work will be released for access worldwide.

Identifier

etd-04142013-195038
https://repository.lsu.edu/gradschool_dissertations/147

OAI identifier oai:identifier

oai:repository.lsu.edu:gradschool_dissertations-1146