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Time-domain finite element methods for Maxwell’s equations in metamaterials. (English) Zbl 1304.78002

Springer Series in Computational Mathematics 43. Berlin: Springer (ISBN 978-3-642-33788-8/hbk; 978-3-642-33789-5/ebook). xii, 302 p. (2013).
Electromagnetic metamaterials are artificially structured composite materials with negative refraction index at certain frequences. Their potentially revolutionary applications stimulated extensive research in physics, electrical engineering, optics, and material science on the construction, design and simulation of efficient metamaterials.
The present monograph provides an introduction to the mathematical and numerical modeling of wave propagation in metamaterials by using time-domain finite element methods for Maxwell’s equations. The book has 9 chapters: 1. Introduction to metamaterials; 2. Introduction to finite element methods; 3. Time-domain finite element methods for metamaterials; 4. Discontinuous Galerkin methods for metamaterials; 5. Superconvergence analysis for metamaterials; 6. A posterioi error estimation; 7. A MATLAB edge element code for metamaterials; 8. Perfectly matched layers; 9. Simulations of wave propagation in metamaterials.
The book is mainly based on results previously published in articles, but it also includes original proofs of superconvergence results. The first chapter gives a brief discussion on the origins of metamaterials and basic electromagnetic properties, describes some metamaterial structures and potential applications. Here the Drude and Lorentz models as governing equations to model wave propagation in metamaterials are introduced. Chapters 2 and 3 contain basic FEM theory, are concerned with the construction and usage of the divergence and curl conforming finite elements, provide existence and uniqueness results for the Drude and Lorentz models and analyze the Crank-Nicholson and leap-frog finite element schemes for solving the time-dependent Maxwell system. In Chapters 4–6 and 8, the authors discuss other important issues for the development of efficient time-domain finite element methods for solving Maxwell’s equations. In Chapter 7, the practical implementation of a mixed finite element method for a 2-D Drude metamaterial model is demonstrated.
The authors provide commented MATLAB source code that may be used for some of the covered topics, or may be modified to solve other problems of interest to the reader. Chapter 9 presents some interesting simulations of wave propagation in metamaterials, for example the demonstration of backward wave propagation, metamaterial electromagnetic cloaking through solving Maxwell’s equations, and solar cell design with metamaterials. Here, also a list of interesting research topics in scientific computing is presented, which could improve computer modeling of metamaterials, thus allowing faster discovery, design and manufacturing of new structures.
The book is rigorous, clearly presented, and written for researchers as well as graduate students in the fields of mathematics, computing, physics, engineering, material sciences, and optics. It helps the reader to get deeper insight and better understanding of actual mathematical and computational problems in the field of metamaterial simulations.

MSC:

78-02 Research exposition (monographs, survey articles) pertaining to optics and electromagnetic theory
65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
65-02 Research exposition (monographs, survey articles) pertaining to numerical analysis
65M15 Error bounds for initial value and initial-boundary value problems involving PDEs
78A25 Electromagnetic theory (general)
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
78A48 Composite media; random media in optics and electromagnetic theory

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