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The book is suitable for advanced undergraduate and beginning graduate students of applied mathematics and engineering. The main theme is the integration of the theory of linear PDEs and the numerical solution of such equations. For each type of PDE, elliptic, parabolic, and hyperbolic, the text contains one chapter on the mathematical theory of the differential equation, followed by one chapter on finite difference methods and one on finite element methods. As preparation, the two-point boundary value problem and the initial-value problem for ODEs are discussed in separate chapters. There is also one chapter on the elliptic eigenvalue problem and eigenfunction expansion. The presentation does not presume a deep knowledge of mathematical and functional analysis. Some background on linear functional analysis and Sobolev spaces, and also on numerical linear algebra, is reviewed in two appendices.

Numerical solutions of differential equations --- Differential equations, Partial --- Equations aux dérivées partielles --- Numerical solutions --- Solutions numériques --- 519.63 --- -Partial differential equations --- Numerical methods for solution of partial differential equations --- Numerical solutions. --- -Numerical methods for solution of partial differential equations --- -519.63 Numerical methods for solution of partial differential equations --- Partial differential equations --- Equations aux dérivées partielles --- Solutions numériques --- 519.63 Numerical methods for solution of partial differential equations --- Numerical analysis --- Mathematical analysis. --- Analysis (Mathematics). --- Numerical analysis. --- Partial differential equations. --- Analysis. --- Numerical Analysis. --- Partial Differential Equations. --- Mathematical analysis --- 517.1 Mathematical analysis --- Differential equations, Partial - Numerical solutions

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    Comprehensive treatment of edge finite element methods 
    Variational theory of Maxwell's equations 
    Error analysis of finite element methods 
    Background material in functional analysis and Sobolev space theory 
    Introduction to inverse problems 

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.


535.13 --- 535.13 Electromagnetic theory (Maxwell) --- Electromagnetic theory (Maxwell) --- finite element method --- computer-aided engineering --- CAE (computer aided engineering) --- Maxwell equations --- Electromagnétisme --- Equations de Maxwell --- Partitial differential equations: domain decomposition methods; elliptic equations; finite difference methods; finite element methods; finite volume methods; hyperbolic equations; inverse problems; iterative solution techniques; methods of lines; multigrid and multilevel methods; parabolic equations; special methods --- Maxwell equations. --- Electromagnétisme --- Electromagnetism --- Finite element method --- Electromagnetics --- Magnetic induction --- Magnetism --- Metamaterials --- Equations, Maxwell --- Differential equations, Partial --- Electromagnetic theory --- FEA (Numerical analysis) --- FEM (Numerical analysis) --- Finite element analysis --- Numerical analysis --- Isogeometric analysis --- Mathematical models --- 519.63 --- 681.3*G18 --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- Numerical solutions of differential equations --- Functional analysis --- eindige elementen --- Finite element method. --- Mathematical models. --- Méthode des éléments finis --- Modèles mathématiques --- Electromagnetism - Mathematical models --- Equations aux derivees partielles --- Equations de maxwell --- Methodes numeriques --- Elements finis

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The numerical treatment of partial differential equations with meshfree discretization techniques has been a very active research area in recent years. Up to now, however, meshfree methods have been in an early experimental stage and were not competitive due to the lack of efficient iterative solvers and numerical quadrature. This volume now presents an efficient parallel implementation of a meshfree method, namely the partition of unity method (PUM). A general numerical integration scheme is presented for the efficient assembly of the stiffness matrix as well as an optimal multilevel solver for the arising linear system. Furthermore, detailed information on the parallel implementation of the method on distributed memory computers is provided and numerical results are presented in two and three space dimensions with linear, higher order and augmented approximation spaces with up to 42 million degrees of freedom.

Differential equations, Elliptic --- Partition of unity method. --- Numerical solutions --- Data processing. --- -681.3*G18 --- 519.63 --- Partition of unity method --- PUM (Numerical analysis) --- Meshfree methods (Numerical analysis) --- 519.63 Numerical methods for solution of partial differential equations --- Numerical methods for solution of partial differential equations --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Elliptic differential equations --- Elliptic partial differential equations --- Linear elliptic differential equations --- Differential equations, Linear --- Differential equations, Partial --- -Data processing --- 681.3*G18 --- Numerical solutions&delete& --- Data processing --- 681.3 *G18 --- Mathematical analysis. --- Analysis (Mathematics). --- Computer mathematics. --- Physics. --- Partial differential equations. --- Applied mathematics. --- Engineering mathematics. --- Analysis. --- Computational Mathematics and Numerical Analysis. --- Numerical and Computational Physics, Simulation. --- Partial Differential Equations. --- Mathematical and Computational Engineering. --- Engineering --- Engineering analysis --- Mathematical analysis --- Partial differential equations --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Dynamics --- Computer mathematics --- Electronic data processing --- Mathematics --- 517.1 Mathematical analysis