Mathematical Foundations of Computational Electromagnetism

“The monograph provides useful mathematical tools for investigations of some problems for electromagnetic fields and their computational realizations.” (Vasil G. Angelov, zbMATH 1452.78001, 2021)“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)

• 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.