Finite Element Methods for Maxwell's Equations

Finite Element Methods For Maxwell's Equations is the first book to present the use of finite elements to analyze Maxwell's equations. This book is part of the Numerical Analysis and Scientific Computation Series.

Peter Monk: Department of Mathematical Sciences University of Delaware Newark, USA

Mathematical models of electromagnetism

1

(14)

Introduction

1

(1)

Maxwell's equations

2

(7)

Constitutive equations for linear media

5

(2)

Interface and boundary conditions

7

(2)

Scattering problems and the radiation condition

9

(3)

Boundary value problems

12

(3)

Time-harmonic problem in a cavity

12

(1)

Cavity resonator

13

(1)

Scattering from a bounded object

13

(1)

Scattering from a buried object

14

(1)

Functional analysis and abstract error estimates

15

(21)

Introduction

15

(1)

Basic functional analysis and the Fredholm alternative

15

(10)

Hilbert space

15

(3)

Linear operators and duality

18

(1)

Variational problems

19

(3)

Compactness and the Fredholm alternative

22

(2)

Hilbert--Schmidt theory of eigenvalues

24

(1)

Abstract finite element convergence theory

25

(11)

Cea's lemma

25

(1)

Discrete mixed problems

26

(6)

Convergence of collectively compact operators

32

(3)

Eigenvalue estimates

35

(1)

Sobolev spaces, vector function spaces and regularity

36

(45)

Introduction

36

(1)

Standard Sobolev spaces

36

(9)

Trace spaces

42

(3)

Regularity results for elliptic equations

45

(3)

Differential operators on a surface

48

(1)

Vector functions with well-defined curl or divergence

49

(12)

Integral identities

50

(2)

Properties of H(div; ω)

52

(3)

Properties of H(curl; ω)

55

(6)

Scalar and vector potentials

61

(4)

The Helmholtz decomposition

65

(4)

A function space for the impedance problem

69

(8)

Curl or divergence conserving transformations

77

(4)

Variational theory for the cavity problem

81

(18)

Introduction

81

(2)

Assumptions on the coefficients and data

83

(1)

The space X and the nullspace of the curl

84

(2)

Helmholtz decomposition

86

(3)

Compactness properties of X0

87

(2)

The variational problem as an operator equation

89

(3)

Uniqueness of the solution

92

(3)

Cavity eigenvalues and resonances

95

(4)

Finite elements on tetrahedra

99

(56)

Introduction

99

(2)

Introduction to finite elements

101

(11)

Sets of polynomials

108

(4)

Meshes and affine maps

112

(6)

Divergence conforming elements

118

(8)

The curl conforming edge elements of Nedelec

126

(17)

Linear edge element

139

(1)

Quadratic edge elements

140

(3)

H1(ω) conforming finite elements

143

(6)

The Clement interpolant

147

(2)

An L2(ω) conforming space

149

(1)

Boundary spaces

150

(5)

Finite elements on hexahedra

155

(11)

Introduction

155

(1)

Divergence conforming elements on hexahedra

155

(3)

Curl conforming hexahedral elements

158

(4)

H1(ω) conforming elements on hexahedra

162

(2)

An L2(ω) conforming space and a boundary space

164

(2)

Finite element methods for the cavity problem

166

(33)

Introduction

166

(2)

Error analysis via duality

168

(8)

The discrete Helmholtz decomposition

170

(1)

Preliminary error analysis

171

(3)

Duality estimate

174

(2)

Error analysis via collective compactness

176

(13)

Pointwise convergence

178

(2)

Collective compactness

180

(8)

Numerical results for the cavity problem

188

(1)

The ellipticized Maxwell system

189

(6)

Discrete ellipticized variational problem

191

(4)

The discrete eigenvalue problem

195

(4)

Topics concerning finite elements

199

(26)

Introduction

199

(3)

The second family of elements on tetrahedra

202

(7)

Divergence conforming element

202

(3)

Curl conforming element

205

(4)

Scalar functions and the de Rham diagram

209

(1)

Curved domains

209

(8)

Locally mapped tetrahedral meshes

210

(4)

Large-element fitting of domains

214

(3)

hp finite elements

217

(8)

H1(ω) conforming hp element

218

(1)

hp curl conforming elements

219

(2)

hp divergence conforming space

221

(1)

de Rham diagram for hp elements

222

(3)

Classical scattering theory

225

(36)

Introduction

225

(1)

Basic integral identities

225

(9)

Scattering by a sphere

234

(14)

Spherical harmonics

236

(2)

Spherical Bessel functions

238

(3)

Series solution of the exterior Maxwell problem

241

(7)

Electromagnetic Calderon operators

248

(6)

The electric-to-magnetic Calderon operator

249

(3)

The magnetic-to-electric Calderon operator

252

(2)

Scattering of a plane wave by a sphere

254

(7)

Uniqueness and Rellich's lemma

254

(2)

Series solution

256

(5)

The scattering problem using Calderon maps

261

(19)

Introduction

261

(1)

Reduction to a bounded domain

262

(2)

Analysis of the reduced problem

264

(10)

Extended Helmholtz decomposition

267

(2)

An operator equation on X0

269

(5)

The discrete problem

274

(6)

Scattering by a bounded inhomogeneity

280

(22)

Introduction

280

(1)

Derivation of the domain-decomposed problem

281

(8)

The finite-dimensional problem

289

(1)

Analysis of the interior finite element problem

290

(8)

Error estimates for the fully discrete problem

298

(4)

Scattering by a buried object

302

(30)

Introduction

302

(1)

Homogeneous isotropic background

303

(12)

Analysis of the scheme

308

(3)

The fully discrete problem

311

(3)

Computational considerations

314

(1)

Perfectly conducting half space

315

(3)

Layered medium

318

(14)

Incident plane waves

318

(3)

The dyadic Green's function

321

(7)

Reduction to a bounded domain

328

(4)

Algorithmic development

332

(62)

Introduction

332

(1)

Solution of the linear system

333

(11)

Phase error in finite element methods

344

(11)

Wavenumber dependent error estimates

345

(6)

Phase error in three dimensional edge elements

351

(4)

A posteriori error estimation

355

(9)

A residual-based error estimator

356

(6)

Numerical experiments

362

(2)

Absorbing boundary conditions

364

(22)

Silver--Muller absorbing boundary condition

365

(5)

Infinite element method

370

(5)

The perfectly matched layer

375

(11)

Far field recovery

386

(8)

Inverse problems

394

(34)

Introduction

394

(3)

The linear sampling method

397

(12)

Implementing the LSM

399

(6)

Numerical results with the LSM

405

(4)

Mathematical aspects of inverse scattering

409

(16)

Uniqueness for the inverse problem

411

(3)

Herglotz wave functions

414

(3)

The far field operators F and B

417

(5)

Mathematical justification of the LSM

422

(3)

Appendices

A Coordinate systems

425

(2)

A.1 Cartesian coordinates

425

(1)

A.2 Spherical coordinates

425

(2)

B. Vector and differential identities

427

(1)

B.1 Vector identities

427

(1)

B.2 Differential identities

427

(1)

B.3 Differential identities on a surface

427

(1)

References

428

(18)

Index

446

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Finite Element Methods for Maxwell's Equations

Summary

Author Biography

Table of Contents

Supplemental Materials