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

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. 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 Since the middle of the last century, computing power has increased sufficiently that the direct numerical approximation of Maxwell's equations is now an increasingly important tool in science and engineering. Parallel to the increasing use of numerical methods in computational electromagnetism there has also been considerable progress in the mathematical understanding of the properties of Maxwell's equations relevant to numerical analysis. The aim of this book is to provide an up to date and sound theoretical foundation for finite element methods in computational electromagnetism. The emphasis is on finite element methods for scattering problems that involve the solution of Maxwell's equations on infinite domains. Suitable variational formulations are developed and justified mathematically. An error analysis of edge finite element methods that are particularly well suited to Maxwell's equations is the main focus of the book. The methods are justified for Lipschitz polyhedral domains that can cause strong singularities in the solution. The book finishes with a short introduction to inverse problems in electromagnetism. "The aim of this book is to provide an up-to-date and sound theoretical foundation for finite element methods in computational electromagnetism. The emphasis in on finite element methods for scattering problems that involve the solution of Maxwell's equations on infinite domains. Suitable variational formulations are developed and justified mathematically. An error analysis of edge finite element methods that are particularly well suited to Maxwell's equations is the main focus of the book. The methods are justified for Lipschitz polyhedral domains that can cause strong singularities in the solution. The book finishes with a short introduction to inverse problems in electromagnetism."--Jacket. This reference provides an up to date and sound theoretical foundation for finite element methods in computational electromagnetism. The emphasis is on finite element methods for scattering problems that involve the solution of Maxwell's equations on infinite domains, and special attention is given to error analysis of edge FEM that are particularly well suited to Maxwell's equations