Title page for ETD etd-06182010-143907


Type of Document Dissertation
Author Cui, Jintao
Author's Email Address cui@math.lsu.edu
URN etd-06182010-143907
Title Multigrid Methods for Maxwell's Equations
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Brenner, Susanne C. Committee Chair
Madden, James Committee Member
Sung, Li-yeng Committee Member
Tom, Michael M. Committee Member
Wan, Xiaoliang Committee Member
Rai, Suresh Dean's Representative
Keywords
  • Maxwell's equations
  • nonconforming finite element methods
  • discontinuous Galerkin methods
  • graded meshes
  • multigrid algorithms
Date of Defense 2010-05-11
Availability unrestricted
Abstract
In this work we study finite element methods for two-dimensional Maxwell's equations and their solutions by multigrid algorithms. We begin with a brief survey of finite element methods for

Maxwell's equations. Then we review the related fundamentals, such as Sobolev spaces, elliptic regularity results, graded meshes, finite element methods for second order problems, and multigrid

algorithms. In Chapter 3, we study two types of nonconforming finite element methods on graded meshes for a two-dimensional curl-curl and grad-div problem that appears in electromagnetics.

The first method is based on a discretization using weakly continuous P1 vector fields. The second method uses discontinuous P1 vector fields. Optimal convergence rates (up

to an arbitrary positive epsilon) in the energy norm and the L2 norm are established for both methods on graded meshes. In Chapter 4, we consider a class of symmetric discontinuous Galerkin methods for a model Poisson problem on graded meshes that share many techniques with the nonconforming methods in Chapter 3. Optimal order error estimates are derived in both the energy norm and the L2 norm. Then we establish the

uniform convergence of W-cycle, V-cycle and F-cycle multigrid algorithms for the resulting discrete problems. In Chapter 5, we propose a new numerical approach for two-dimensional Maxwell's equations that is based on the 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. We illustrate this new approach by a P1 finite element method. In Chapter 6, we first report numerical results for multigrid

algorithms applied to the discretized curl-curl and grad-div problem using nonconforming finite element methods. Then we present multigrid results for Maxwell's equations based on

the approach introduced in Chapter 5. All the theoretical results obtained in this dissertation are confirmed by numerical experiments.

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