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This textbook is devoted to second order linear partial differential equations. The focus is on variational formulations in Hilbert spaces. It contains elliptic equations, including some basic results on Fredholm alternative and spectral theory, some useful notes on functional analysis, a brief presentation of Sobolev spaces and their properties, saddle point problems, parabolic equations and hyperbolic equations. Many exercises are added, and the complete solution of all of them is included. The work is mainly addressed to students in Mathematics, but also students in Engineering with a good mathematical background should be able to follow the theory presented here.

Functional analysis --- Partial differential equations --- Mathematics --- Computer. Automation --- differentiaalvergelijkingen --- functies (wiskunde) --- computers

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Numerical solutions of differential equations --- Partial differential equations --- Décomposition (méthode mathématique) --- Équations aux dérivées partielles --- Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- 681.3 *G18 Partial differential equations: difference methods; elliptic equations; finite element methods; hyperbolic equations; method of lines; parabolic equations (Numerical analysis) --- Decomposition method --- Differential equations, Partial --- 519.6 --- 681.3 *G18 --- 519.6 Computational mathematics. Numerical analysis. Computer programming --- Computational mathematics. Numerical analysis. Computer programming --- Method, Decomposition --- Operations research --- Programming (Mathematics) --- System analysis --- Numerical analysis --- Numerical solutions --- Decomposition method. --- Numerical solutions. --- Solutions numériques --- Solutions numériques.

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This book deals with the mathematical analysis and the numerical approximation of time-harmonic eddy current problems. It is self-contained and suitable for mathematicians and engineers working in the field, and also accessible for beginners. Depending on the choice of the physical unknowns, these problems are formulated in different variational ways, with specific attention to the topology of the computational domain. Finite elements of nodal or edge type are used for numerical approximation, and a complete analysis of convergence is performed. A specific feature of the book is the emphasis given to saddle-point formulations in terms of the magnetic and electric fields. New results for voltage or current intensity excitation problems are also presented.

Partial differential equations --- Mathematical analysis --- Mathematics --- Planning (firm) --- Computer. Automation --- differentiaalvergelijkingen --- analyse (wiskunde) --- computers --- informatica --- mathematische modellen --- wiskunde --- Eddy currents (Electric) --- Harmonic analysis --- Time-series analysis --- 535.13 --- 535.13 Electromagnetic theory (Maxwell) --- Electromagnetic theory (Maxwell) --- Analysis of time series --- Autocorrelation (Statistics) --- Mathematical statistics --- Probabilities --- Analysis (Mathematics) --- Functions, Potential --- Potential functions --- Banach algebras --- Calculus --- Bessel functions --- Fourier series --- Harmonic functions --- Electric currents --- Mathematical models

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This book deals with the mathematical analysis and the numerical approximation of time-harmonic eddy current problems. It is self-contained and suitable for mathematicians and engineers working in the field, and also accessible for beginners. Depending on the choice of the physical unknowns, these problems are formulated in different variational ways, with specific attention to the topology of the computational domain. Finite elements of nodal or edge type are used for numerical approximation, and a complete analysis of convergence is performed. A specific feature of the book is the emphasis given to saddle-point formulations in terms of the magnetic and electric fields. New results for voltage or current intensity excitation problems are also presented.

Partial differential equations --- Mathematical analysis --- Mathematics --- Planning (firm) --- Computer. Automation --- differentiaalvergelijkingen --- analyse (wiskunde) --- computers --- informatica --- mathematische modellen --- wiskunde

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Presenting topics that have not previously been contained in a single volume, this book offers an up-to-date review of computational methods in electromagnetism, with a focus on recent results in the numerical simulation of real-life electromagnetic problems and on theoretical results that are useful in devising and analyzing approximation algorithms. Based on four courses delivered in Cetraro in June 2014, the material covered includes the spatial discretization of Maxwell’s equations in a bounded domain, the numerical approximation of the eddy current model in harmonic regime, the time domain integral equation method (with an emphasis on the electric-field integral equation) and an overview of qualitative methods for inverse electromagnetic scattering problems. Assuming some knowledge of the variational formulation of PDEs and of finite element/boundary element methods, the book is suitable for PhD students and researchers interested in numerical approximation of partial differential equations and scientific computing.

Mathematics. --- Computational Mathematics and Numerical Analysis. --- Optics and Electrodynamics. --- Partial Differential Equations. --- Numerical and Computational Physics. --- Differential equations, partial. --- Computer science --- Mathématiques --- Informatique --- Mathematics - General --- Mathematics --- Physical Sciences & Mathematics --- Partial differential equations. --- Computer mathematics. --- Physics. --- Optics. --- Electrodynamics. --- Dynamics --- Physics --- Light --- Natural philosophy --- Philosophy, Natural --- Physical sciences --- Computer mathematics --- Discrete mathematics --- Electronic data processing --- Partial differential equations --- Math --- Science --- Classical Electrodynamics. --- Numerical and Computational Physics, Simulation. --- Electromagnetism --- Computer simulation. --- Electromagnetics --- Magnetic induction --- Magnetism --- Metamaterials