Finite Element Methods for Maxwell's Equations
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. The main focus of the book is an error analysis of edge finite element methods that are particularly well suited to Maxwell's equations. The methods are justified for Lipschitz polyhedral domains that can cause strong singularities in the solution. The book is divided into 14 chapters. After the preparatory work in Chapters 2 and 3 (functional analysis and abstract error estimates, Sobolev spaces and vector function spaces), Chapter 4 contains a discussion of a simple model problem for Maxwell's equations. Chapters 5 and 6 form a central part of the book. They present a detailed description of the original Nédélec finite element spaces. In Chapter 7, the author gives the finite element discretization of the cavity problem and presents two convergence proofs for this method. Chapter 8 presents basic topics concerning finite elements and gives a brief introduction to hp finite element methods for Maxwell's equations. A central task of computational electromagnetism is the approximation of scattering problems. The presentation of these problems starts in Chapter 9, with classical scattering by a sphere, where the author derives the famous integral representation of the solution to Maxwell's equations called the Stratton-Chu formula. In addition, the author derives classical series representations of the solution of Maxwell's equations. These are used in Chapter 10 to derive a semi-discrete method for the scattering problem utilizing the electromagnetic equivalent of the Dirichlet to Neumann map. A fully discrete domain-decomposed version of this algorithm is proposed and analysed in Chapter 11. In Chapter 12, the author studies a coupled integral equation and finite element method due to Hazard, Lenoir and Cutzach. Chapter 13 is devoted to a discussion of some issues related to practical aspects of solving Maxwell's equations. The final Chapter gives a short introduction to inverse problems in electromagnetism. The book will be of interest to mathematicians, physicists and researchers concerned with Maxwell's equations. The three chapters, 4, 5 and 7, could be useful in a graduate course on finite element methods.