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

In light of increasing uses for direct numerical approximations of Maxwell's equations in science and engineering, this text provides mathematics graduate students and researchers with a theoretical foundation for finite element methods in computational electromagnetism. Monk (mathematical sciences, U. of Delaware) emphasizes finite element methods for scattering problems involving the solutions of Maxwell's equations on infinite domains. The book's main focus is on an error analysis of edge finite element methods that are well suited to Maxwell's equations. The book concludes with a short introduction to inverse problems in electromagnetism. Front cover......Page 1 Numerical Mathematics and Scientific Computation Series......Page 2 Title page......Page 3 Date-line......Page 4 Preface......Page 5 CONTENTS......Page 11 1.1 Introduction......Page 15 1.2 Maxwell's equations......Page 16 1.2.1 Constitutive equations for linear media......Page 19 1.2.2 Interface and boundary conditions......Page 21 1.3 Scattering problems and the radiation condition......Page 23 1.4.1 Time-harmonic problem in a cavity......Page 26 1.4.3 Scattering from a bounded object......Page 27 1.4.4 Scattering from a buried object......Page 28 2.2.1 Hilbert space......Page 29 2.2.2 Linear operators and duality......Page 32 2.2.3 Variational problems......Page 33 2.2.4 Compactness and the Fredholm alternative......Page 36 2.2.5 Hilbert-Schmidt theory of eigenvalues......Page 38 2.3.1 Cea's lemma......Page 39 2.3.2 Discrete mixed problems......Page 40 2.3.3 Convergence of collectively compact operators......Page 46 2.3.4 Eigenvalue estimates......Page 49 3.2 Standard Sobolev spaces......Page 50 3.2.1 Trace spaces......Page 56 3.3 Regularity results for elliptic equations......Page 59 3.4 Differential operators on a surface......Page 62 3.5 Vector functions with well-defined curl or divergence......Page 63 3.5.1 Integral identities......Page 64 3.5.2 Properties of $H(\div;\Omega)$......Page 66 3.5.3 Properties of $H(\curl;\Omega)$......Page 69 3.6 Scalar and vector potentials......Page 75 3.7 The Helmholtz decomposition......Page 79 3.8 A function space for the impedance problem......Page 83 3.9 Curl or divergence conserving transformations......Page 91 4.1 Introduction......Page 95 4.2 Assumptions on the coefficients and data......Page 97 4.3 The space $X$ and the nullspace of the curl......Page 98 4.4 Helmholtz decomposition......Page 100 4.4.1 Compactness properties of $X_0$......Page 101 4.5 The variational problem as an operator equation......Page 103 4.6 Uniqueness of the solution......Page 106 4.7 Cavity eigenvalues and resonances......Page 109 5.1 Introduction......Page 113 5.2 Introduction to finite elements......Page 115 5.2.1 Sets of polynomials......Page 122 5.3 Meshes and affine maps......Page 126 5.4 Divergence conforming elements......Page 132 5.5 The curl conforming edge elements of Nedelec......Page 140 5.5.1 Linear edge element......Page 153 5.5.2 Quadratic edge elements......Page 154 5.6 $H^1(\Omega)$ conforming finite elements......Page 157 5.6.1 The Clement interpolant......Page 161 5.7 An $L^2(\Omega)$ conforming space......Page 163 5.8 Boundary spaces......Page 164 6.2 Divergence conforming elements on hexahedra......Page 169 6.3 Curl conforming hexahedral elements......Page 172 6.4 $H^1(\Omega)$ conforming elements on hexahedra......Page 176 6.5 An $L^2(\Omega)$ conforming space and a boundary space......Page 178 7.1 Introduction......Page 180 7.2 Error analysis via duality......Page 182 7.2.1 The discrete Helmholtz decomposition......Page 184 7.2.2 Preliminary error analysis......Page 185 7.2.3 Duality estimate......Page 188 7.3 Error analysis via collective compactness......Page 190 7.3.1 Point wise convergence......Page 192 7.3.2 Collective compactness......Page 194 7.3.3 Numerical results for the cavity problem......Page 202 7.4 The ellipticized Maxwell system......Page 203 7.4.1 Discrete ellipticized variational problem......Page 205 7.5 The discrete eigenvalue problem......Page 209 8.1 Introduction......Page 213 8.2.1 Divergence conforming element......Page 216 8.2.2 Curl conforming element......Page 219 8.3 Curved domains......Page 223 8.3.1 Locally mapped tetrahedral meshes......Page 224 8.3.2 Large-element fitting of domains......Page 228 8.4 hp finite elements......Page 231 8.4.1 $H^1(\Omega)$ conforming $hp$ element......Page 232 8.4.2 $hp$ curl conforming elements......Page 233 8.4.3 $hp$ divergence conforming space......Page 235 8.4.4 de Rham diagram for $hp$ elements......Page 236 9.2 Basic integral identities......Page 239 9.3 Scattering by a sphere......Page 248 9.3.1 Spherical harmonics......Page 250 9.3.2 Spherical Bessel functions......Page 252 9.3.3 Series solution of the exterior Maxwell problem......Page 255 9.4 Electromagnetic Calderon operators......Page 262 9.4.1 The electric-to-magnetic Calderon operator......Page 263 9.4.2 The magnetic-to-electric Calderon operator......Page 266 9.5.1 Uniqueness and Rellich's lemma......Page 268 9.5.2 Series solution......Page 270 10.1 Introduction......Page 275 10.2 Reduction to a bounded domain......Page 276 10.3 Analysis of the reduced problem......Page 278 10.3.1 Extended Hclmholtz decomposition......Page 281 10.3.2 An operator equation on $\tilde{X}_0$......Page 283 10.4 The discrete problem......Page 288 11.1 Introduction......Page 294 11.2 Derivation of the domain-decomposed problem......Page 295 11.3 The finite-dimensional problem......Page 303 11.4 Analysis of the interior finite element problem......Page 304 11.5 Error estimates for the fully discrete problem......Page 312 12.1 Introduction......Page 316 12.2 Homogeneous isotropic background......Page 317 12.2.1 Analysis of the scheme......Page 322 12.2.2 The fully discrete problem......Page 325 12.2.3 Computational considerations......Page 328 12.3 Perfectly conducting half space......Page 329 12.4.1 Incident plane waves......Page 332 12.4.2 The dyadic Green's function......Page 335 12.4.3 Reduction to a bounded domain......Page 342 13.1 Introduction......Page 346 13.2 Solution of the linear system......Page 347 13.3 Phase error in finite element methods......Page 358 13.3.1 Wavenumber dependent error estimates......Page 359 13.3.2 Phase error in three dimensional edge elements......Page 365 13.4 A posteriori error estimation......Page 369 13.4.1 A residual-based error estimator......Page 370 13.4.2 Numerical experiments......Page 376 13.5 Absorbing boundary conditions......Page 378 13.5.1 Silver-Miiller absorbing boundary condition......Page 379 13.5.2 Infinite element method......Page 384 13.5.3 The perfectly matched layer......Page 389 13.6 Far field recovery......Page 400 14.1 Introduction......Page 408 14.2 The linear sampling method......Page 411 14.2.1 Implementing the LSM......Page 413 14.2.2 Numerical results with the LSM......Page 419 14.3 Mathematical aspects of inverse scattering......Page 423 14.3.1 Uniqueness for the inverse problem......Page 425 14.3.2 Herglotz wave functions......Page 428 14.3.3 The far field operators $\mathbf{F}$ and $\bold{\mathcal{B}}$......Page 431 14.3.4 Mathematical justification of the LSM......Page 436 A.2 Spherical coordinates......Page 439 B.3 Differential identities on a surface......Page 441 References......Page 442 Index......Page 460 Back cover......Page 465 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 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.