Finite Element Methods for Maxwell's Equations

by Monk, Peter

• ISBN13: 9780198508885
• ISBN10: 0198508883
• Format: Hardcover
• Copyright: 2003-06-19
• Publisher: Clarendon Press

Summary

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.

Author Biography

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

Table of Contents

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