Mathematics

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 analysis involves a complete justification of the discrete de Rham diagram and discrete compactness of edge elements. The numerical methods are justified for Lipschitz polyhedral domains that can cause strong singularities in the solution. The book ends with a short introduction to inverse problems in electromagnetism.

The main purpose of the book is to introduce the numerical integration of the Cauchy problem for delay differential equations (DDEs) and of the neutral type. Comparisons between DDEs and ordinary differential equations (ODEs) are made using examples illustrating some unexpected and often surprising behaviours of the true and numerical solutions. The book briefly reviews the various approaches existing in the literature and develops an error and well-posedness analysis for general one-step and multistep methods. The continuous extensions of Runge-Kutta methods are presented in detail, which are useful for more general problems such as dense output and discontinuous equations. Some deeper insight into convergence and superconvergence is then carried out for DDEs with various kinds of delays. The stepsize control mechanism is developed on a firm mathematical basis. Classical results and an unconventional analysis of stability with respect to forcing term are reviewed for ODEs in view of the subsequent stability analysis for DDEs. Moreover, an exhaustive description of stability domains for some test DDEs is carried out and the corresponding investigations for the numerical methods are made. Reformulations of DDEs as partial differential equations and subsequent semi-discretization are described and compared with the classical approach. A list of available codes is provided.

The book deals with the numerical solution of structured Markov chains which include M/G/1 and G/M/1-type Markov chains, QBD processes, non-skip-free queues, and tree-like stochastic processes and has a wide applicability in queueing theory and stochastic modeling. It presents in a unified language the most up to date algorithms, which are so far scattered in diverse papers, written with different languages and notation. It contains a thorough treatment of numerical algorithms to solve these problems, from the simplest to the most advanced and most efficient. Nonlinear matrix equations are at the heart of the analysis of structured Markov chains, they are analysed both from the theoretical, from the probabilistic, and from the computational point of view. The set of methods for solution contains functional iterations, doubling methods, logarithmic reduction, cyclic reduction, and subspace iteration, all are described and analysed in detail. They are also adapted to interesting specific queueing models coming from applications. The book also offers a comprehensive and self-contained treatment of the structured matrix tools which are at the basis of the fastest algorithmic techniques for structured Markov chains. Results about Toeplitz matrices, displacement operators, and Wiener-Hopf factorizations are reported to the extent that they are useful for the numerical treatment of Markov chains. Every and all solution methods are reported in detailed algorithmic form so that they can be coded in a high-level language with minimum effort.

Spectral methods have long been popular in direct and large eddy simulation of turbulent flows, but their use in areas with complex-geometry computational domains has historically been much more limited. More recently, the need to find accurate solutions to the viscous flow equations around complex configurations has led to the development of high-order discretization procedures on unstructured meshes, which are also recognized as more efficient for solution of time-dependent oscillatory solutions over long time periods. This book, an updated edition on the original text, presents the recent and significant progress in multi-domain spectral methods at both the fundamental and application level. Containing material on discontinuous Galerkin methods, non-tensorial nodal spectral element methods in simplex domains, and stabilization and filtering techniques, this text introduces the use of spectral/hp element methods with particular emphasis on their application to unstructured meshes. It provides a detailed explanation of the key concepts underlying the methods along with practical examples of their derivation and application.