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Discretization of electromagnetic problems: the “generalized finite differences” approach. (English) Zbl 1109.78001
Schilders, W. H. A. (ed.) et al., Handbook of numerical analysis. Vol XIII. Special volume: Numerical methods in electromagnetics. Amsterdam: Elsevier/North Holland (ISBN 0-444-51375-2/hbk). Handbook of Numerical Analysis 13, 105-197 (2005).

Basically, this paper is a monograph, rather than a paper. It is divided into four chapters:

The first chapter introduces the mathematical basis necessary. The topics covered include ideas such as affine space, manifolds and the idea of orientation which lead on to the idea of chains and similar quantities. The importance of boundaries and the ideas of metrics also have their place. The second chapter gives an analysis of Maxwell’s equations, expressed in terms of differential forms, rather than vector fields. An expansion of differential forms is given and the simplicity of Stokes’ theorem in this notation is pointed out and the idea of a magnetic field as a 2-form is discussed. This leads on to the appropriate expression of the Faraday and Ampère laws with a further analysis leading to the Hodge operator. Another topic discussed is the wedge product which leads to expressions for various energies the electromagnetic field and the Poynting theorem.

In the third chapter the author passes on to the idea of discretization. He opens by treating a model which may be used as a paradigm for a number of problems. He passes on to discuss the ideas of primal and dual modes. He then considers the details of the discretization processes and the network equations. The discretization processes for Maxwell’s equations are discussed and attention is drawn to the possibility of the so-called “ghost modes”. The fourth and final chapter discusses the use of finite elements. There is a treatment of consistency and stability and an indication of how the ideas can be applied to time-dependent problems. The application of Whitney and higher degree forms is considered.

The monograph forms a scholarly account of the field but the treament is highly abstract and it is not clear why a treatment by differential forms is better than a treatment by vectors. A comparison of some problem in both maps would be of interest. The paper closes with a list of 100 references and there is a list of suggestions for further reading.

MSC:
78-02Research monographs (optics, electromagnetic theory)
78M25Numerical methods in optics
78M10Finite element methods (optics)
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
65M06Finite difference methods (IVP of PDE)
58A10Differential forms (global analysis)
35Q60PDEs in connection with optics and electromagnetic theory