Finite Element Methods For Maxwell's Equations is the first book to present the use of finite elements to analyse Maxwell's equations. This book is part of the Numerical Analysis and Scientific Computation Series.

An analysis of the three-dimensional (3-D) finite element formulation of Maxwell's equations governing classical electromagnetic propagation in dielectrics is given including its analogy to Navier's equation. The weak form of the electric field equation is reviewed along with dispersion analysis and approximation equations. Radiation boundary conditions are also explored to include paraxial absorber, Sandier absorber, and other absorber comparisons. In addition, time domains vs. frequency domains are investigated with a listing of possible advantages and disadvantages. It was concluded that if large-scale calculations need to be done today, time-domain techniques provide the most practicable means; however, it is still premature to promote such solvers as production level tools for engineers.

In this dissertation we are concerned with the analysis of the finite element method for the time-dependent Maxwell interface problem when Nedelec and Raviart-Thomas finite elements are employed and preconditioning of the resulting linear system when implicit time schemes are used. We first investigate the finite element method proposed by Makridakis and Monk in 1995. After studying the regularity of the solution to time dependent Maxwell's problem and providing approximation estimates for the Fortin operator, we are able to give the optimal error estimate for the semi-discrete scheme for Maxwell's equations. Then we study preconditioners for linear systems arising in the finite element method for time-dependent Maxwell's equations using implicit time-stepping. Such linear systems are usually very large but sparse and can only be solved iteratively. We consider overlapping Schwarz methods and multigrid methods and extend some existing theoretical convergence results. For overlapping Schwarz methods, we provide numerical experiments to confirm the theoretical analysis.

This book contains detailed lecture notes on four topics at the forefront of current research in computational mathematics. Each set of notes presents a self-contained guide to a current research area and has an extensive bibliography. In addition, most of the notes contain detailed proofs of the key results. The notes start from a level suitable for first year graduate students in applied mathematics, mathematical analysis or numerical analysis, and proceed to current research topics. The reader should therefore be able to gain quickly an insight into the important results and techniques in each area without recourse to the large research literature. Current (unsolved) problems are also described and directions for future research are given. This book is also suitable for professional mathematicians who require a succint and accurate account of recent research in areas parallel to their own, and graduates in mathematical sciences.

This book explores the connection between algebraic structures in topology and computational methods for 3-dimensional electric and magnetic field computation. The connection between topology and electromagnetism has been known since the 19th century, but there has been little exposition of its relevance to computational methods in modern topological language. This book is an effort to close that gap. It will be of interest to people working in finite element methods for electromagnetic computation and those who have an interest in numerical and industrial applications of algebraic topology.

Since their emergence, finite element methods have taken a place as one of the most versatile and powerful methodologies for the approximate numerical solution of Partial Differential Equations. These methods are used in incompressible fluid flow, heat, transfer, and other problems. This book provides researchers and practitioners with a concise guide to the theory and practice of least-square finite element methods, their strengths and weaknesses, established successes, and open problems.

This book deals with the analysis and development of numerical methods for the time-domain analysis of multiphysical effects in superconducting circuits of particle accelerator magnets. An important challenge is the simulation of “quenching”, i.e. the transition of a material from the superconducting to the normally electrically conductive state. The book analyses complex mathematical structures and presents models to simulate such quenching events in the context of generalized circuit elements. Furthermore, it proposes efficient parallelized algorithms with guaranteed convergence properties for the simulation of multiphysical problems. Spanning from theoretical concepts to applied research, and featuring rigorous mathematical presentations on one side, as well as simplified explanations of many complex issues, on the other side, this book provides graduate students and researchers with a comprehensive introduction on the state of the art and a source of inspiration for future research. Moreover, the proposed concepts and methods can be extended to the simulation of multiphysical phenomena in different application contexts.

This book presents a comprehensive mathematical approach for solving stochastic magnetic field problems. It discusses variability in material properties and geometry, with an emphasis on the preservation of structural physical and mathematical properties. It especially addresses uncertainties in the computer simulation of magnetic fields originating from the manufacturing process. Uncertainties are quantified by approximating a stochastic reformulation of the governing partial differential equation, demonstrating how statistics of physical quantities of interest, such as Fourier harmonics in accelerator magnets, can be used to achieve robust designs. The book covers a number of key methods and results such as: a stochastic model of the geometry and material properties of magnetic devices based on measurement data; a detailed description of numerical algorithms based on sensitivities or on a higher-order collocation; an analysis of convergence and efficiency; and the application of the developed model and algorithms to uncertainty quantification in the complex magnet systems used in particle accelerators.