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Mathematical Foundations of Computational Electromagnetism


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Table of Contents

Foreword.- Physical framework and models.- Electromagnetic fields and Maxwell's equations.- Stationary equations.- Coupling with other models.- Approximate models.- Elements of mathematical classifications.- Boundary conditions and radiation conditions.- Energy matters.- Bibliographical notes.- Basic applied functional analysis.- Function spaces for scalar fields.- Vector fields: standard function spaces.- Practical function spaces in the (t, x) variable.- Complements of applied functional analysis.- Vector fields: tangential trace revisited.- Scalar and vector potentials: the analyst's and topologist's points of view.- Extraction of scalar potentials and consequences.- Extraction of vector potentials.- Extraction of vector potentials - Vanishing normal trace.- Extraction of vector potentials - Complements.- Helmholtz decompositions.- Abstract mathematical framework.- Basic Results.- Static problems.- Time-dependent problems.- Time-dependent problems: improved regularity results.- Time-harmonic problems.- Summing up.- Analyses of exact problems: first-order models.- Energy matters: uniqueness of the fields.- Well-posedness.- Analyses of approximate models.- Electrostatic problem.- Magnetostatic problem.- Further comments around static problems.- Other approximate models.- Analyses of exact problems: second-order models.- First-order to second-order equations.- Well-posedness of the second-order Maxwell equations.- Second-order to first-order equations.- Other variational formulations.- Compact imbeddings.- Improved regularity for augmented and mixed augmented formulations.- Analyses of time-harmonic problems.- Compact imbeddings: complements.- Free vibrations in a domain encased in a cavity.- Sustained vibrations.- Interface problem between a dielectric and a Lorentz material.- Comments.- Dimensionally reduced models: derivation and analyses.- Two-and-a-half dimensional (2 1/2 2D) models.- Two-dimensional (2D) models.- Some results of functional analysis.- Existence and uniqueness results (2D problems).- Analyses of coupled models.- The Vlasov-Maxwell and Vlasov-Poisson systems.- Magnetohydrodynamics.- References.- Index of function spaces.- Basic Spaces.- Electromagnetic spaces.- Dimension reduction and weighted spaces.- Spaces measuring time regularity.- List of Figures.- Index.


"This book presents an in-depth treatment of various mathematical aspects of electromagnetism and Maxwell's equations. ... The text is entirely self-contained, assuming from the reader only an undergraduate-level background in analysis. It is suitable for students and researchers in applied mathematics interested in Maxwell's equations and their approximate or coupled models." (Agustin Martin, Mathematical Reviews, January, 2019)

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