Finite Element Methods for Maxwell's Equations (Numerical Analysis and Scientific Computation Series)

0198508883......Page 1
Contents......Page 12
1.1 Introduction......Page 16
1.2 Maxwell's equations......Page 17
1.2.1 Constitutive equations for linear media......Page 20
1.2.2 Interface and boundary conditions......Page 22
1.3 Scattering problems and the radiation condition......Page 24
1.4.1 Time-harmonic problem in a cavity......Page 27
1.4.3 Scattering from a bounded object......Page 28
1.4.4 Scattering from a buried object......Page 29
2.2.1 Hilbert space......Page 30
2.2.2 Linear operators and duality......Page 33
2.2.3 Variational problems......Page 34
2.2.4 Compactness and the Fredholm alternative......Page 37
2.2.5 Hilbert-Schmidt theory of eigenvalues......Page 39
2.3.1 Cea's lemma......Page 40
2.3.2 Discrete mixed problems......Page 41
2.3.3 Convergence of collectively compact operators......Page 47
2.3.4 Eigenvalue estimates......Page 50
3.2 Standard Sobolev spaces......Page 51
3.2.1 Trace spaces......Page 57
3.3 Regularity results for elliptic equations......Page 60
3.4 Differential operators on a surface......Page 63
3.5 Vector functions with well-defined curl or divergence......Page 64
3.5.1 Integral identities......Page 65
3.5.2 Properties of H(div;Ω)......Page 67
3.5.3 Properties of H(curl;Ω)......Page 70
3.6 Scalar and vector potentials......Page 76
3.7 The Helmholtz decomposition......Page 80
3.8 A function space for the impedance problem......Page 84
3.9 Curl or divergence conserving transformations......Page 92
4.1 Introduction......Page 96
4.2 Assumptions on the coefficients and data......Page 98
4.3 The space X and the nullspace of the curl......Page 99
4.4 Helmholtz decomposition......Page 101
4.4.1 Compactness properties of X[sub(0)]......Page 102
4.5 The variational problem as an operator equation......Page 104
4.6 Uniqueness of the solution......Page 107
4.7 Cavity eigenvalues and resonances......Page 110
5.1 Introduction......Page 114
5.2 Introduction to finite elements......Page 116
5.2.1 Sets of polynomials......Page 123
5.3 Meshes and affine maps......Page 127
5.4 Divergence conforming elements......Page 133
5.5 The curl conforming edge elements of Nédélec......Page 141
5.5.1 Linear edge element......Page 154
5.5.2 Quadratic edge elements......Page 155
5.6H[sup(1)](Ω) conforming finite elements......Page 158
5.6.1 The Clément interpolant......Page 162
5.7 An L[sup(2)](Ω) conforming space......Page 164
5.8 Boundary spaces......Page 165
6.2 Divergence conforming elements on hexahedra......Page 170
6.3 Curl conforming hexahedral elements......Page 173
6.4H[sup(1)](Ω) conforming elements on hexahedra......Page 177
6.5 An L[sup(2)](Ω) conforming space and a boundary space......Page 179
7.1 Introduction......Page 181
7.2 Error analysis via duality......Page 183
7.2.1 The discrete Helmholtz decomposition......Page 185
7.2.2 Preliminary error analysis......Page 186
7.2.3 Duality estimate......Page 189
7.3 Error analysis via collective compactness......Page 191
7.3.1 Pointwise convergence......Page 193
7.3.2 Collective compactness......Page 195
7.3.3 Numerical results for the cavity problem......Page 203
7.4 The ellipticized Maxwell system......Page 204
7.4.1 Discrete ellipticized variational problem......Page 206
7.5 The discrete eigenvalue problem......Page 210
8.1 Introduction......Page 214
8.2.1 Divergence conforming element......Page 217
8.2.2 Curl conforming element......Page 220
8.3 Curved domains......Page 224
8.3.1 Locally mapped tetrahedral meshes......Page 225
8.3.2 Large-element fitting of domains......Page 229
8.4hp finite elements......Page 232
8.4.1H[sup(1)](Ω) conforming hp element......Page 233
8.4.2hp curl conforming elements......Page 234
8.4.3hp divergence conforming space......Page 236
8.4.4 de Rham diagram for hp elements......Page 237
9.2 Basic integral identities......Page 240
9.3 Scattering by a sphere......Page 249
9.3.1 Spherical harmonics......Page 251
9.3.2 Spherical Bessel functions......Page 253
9.3.3 Series solution of the exterior Maxwell problem......Page 256
9.4 Electromagnetic Calderon operators......Page 263
9.4.1 The electric-to-magnetic Calderon operator......Page 264
9.4.2 The magnetic-to-electric Calderon operator......Page 267
9.5.1 Uniqueness and Rellich's lemma......Page 269
9.5.2 Series solution......Page 271
10.1 Introduction......Page 276
10.2 Reduction to a bounded domain......Page 277
10.3 Analysis of the reduced problem......Page 279
10.3.1 Extended Helmholtz decomposition......Page 282
10.3.2 An operator equation on X[sub(0)]......Page 284
10.4 The discrete problem......Page 289
11.1 Introduction......Page 295
11.2 Derivation of the domain-decomposed problem......Page 296
11.3 The finite-dimensional problem......Page 304
11.4 Analysis of the interior finite element problem......Page 305
11.5 Error estimates for the fully discrete problem......Page 313
12.1 Introduction......Page 317
12.2 Homogeneous isotropic background......Page 318
12.2.1 Analysis of the scheme......Page 323
12.2.2 The fully discrete problem......Page 326
12.2.3 Computational considerations......Page 329
12.3 Perfectly conducting half space......Page 330
12.4.1 Incident plane waves......Page 333
12.4.2 The dyadic Green's function......Page 336
12.4.3 Reduction to a bounded domain......Page 343
13.1 Introduction......Page 347
13.2 Solution of the linear system......Page 348
13.3 Phase error in finite element methods......Page 359
13.3.1 Wavenumber dependent error estimates......Page 360
13.3.2 Phase error in three dimensional edge elements......Page 366
13.4A posteriori error estimation......Page 370
13.4.1 A residual-based error estimator......Page 371
13.4.2 Numerical experiments......Page 377
13.5 Absorbing boundary conditions......Page 379
13.5.1 Silver-Müller absorbing boundary condition......Page 380
13.5.2 Infinite element method......Page 385
13.5.3 The perfectly matched layer......Page 390
13.6 Far field recovery......Page 401
14.1 Introduction......Page 409
14.2 The linear sampling method......Page 412
14.2.1 Implementing the LSM......Page 414
14.2.2 Numerical results with the LSM......Page 420
14.3 Mathematical aspects of inverse scattering......Page 424
14.3.1 Uniqueness for the inverse problem......Page 426
14.3.2 Herglotz wave functions......Page 429
14.3.3 The far field operators F and B......Page 432
14.3.4 Mathematical justification of the LSM......Page 437
B.3 Differential identities on a surface......Page 442
A.2 Spherical coordinates......Page 440
References......Page 443
Index......Page 461