Mathematical Foundations of Computational Electromagnetism

This book presents an in-depth treatment of various mathematical aspects of electromagnetism and Maxwell's equations: from modeling issues to well-posedness results and the coupled models of plasma physics (Vlasov-Maxwell and Vlasov-Poisson systems) and magnetohydrodynamics (MHD). These equations and boundary conditions are discussed, including a brief review of absorbing boundary conditions. The focus then moves to well‐posedness results. The relevant function spaces are introduced, with an emphasis on boundary and topological conditions. General variational frameworks are defined for static and quasi-static problems, time-harmonic problems (including fixed frequency or Helmholtz-like problems and unknown frequency or eigenvalue problems), and time-dependent problems, with or without constraints. They are then applied to prove the well-posedness of Maxwell’s equations and their simplified models, in the various settings described above. The book is completed with a discussion of dimensionally reduced models in prismatic and axisymmetric geometries, and a survey of existence and uniqueness results for the Vlasov-Poisson, Vlasov-Maxwell and MHD equations.

The book addresses mainly researchers in applied mathematics who work on Maxwell’s equations. However, it can be used for master or doctorate-level courses on mathematical electromagnetism as it requires only a bachelor-level knowledge of analysis.

• Maxwell equations
• Vlasov Maxwell model
• Vlasov Maxwell equation
• Vlasov Maxwell system
• Magnetohydrodynamics
• electromagnetic Darwin model
• Helmholtz decomposition
• electrostatic problem
• magnetostatics
• magnetostatic field problem
• time harmonic electromagnetic fields
• Vlasov Poisson systems
• Vlasov Poisson equation
• linear Vlasov equation

“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)

• Department of Mathematics, Ariel University, Ariel, Israel

  Franck Assous

• ParisTech, ENSTA, Palaiseau, France

  Patrick Ciarlet

• Université de Lorraine, Institut Élie Cartan de Lorraine, Vandoeuvre-lès-Nancy, France

  Simon Labrunie