Overview

Continuamentenasconoifatti 1 aconfusionedelleteorie 2 Carlo Dossi Electromagnetism is withoutany doubt a fascinating area of physics, engineering and mathematics. Since the early pioneeringworks ofAmpere, Faraday, and Maxwell, the scienti?cliteratureon this subject has become immense, and books devoted to almost all of its aspects have been published in the meantime. However, webelievethatthereisstillsomeplacefornew booksdealingwithel- tromagnetism, particularly if they are focused on more speci?c models, or try to mix different levels of analysis: rigorous mathematical results, sound numerical appro- mation schemes, real-life examples from physics and engineering. The complete mathematical description of electromagnetic problems is provided by the celebrated Maxwell equations, a system of partial differential equations - pressed interms ofphysical quantitiesliketheelectric?eld, themagnetic?eld and the currentdensity.Maxwell'scontributiontotheformulationofthese equationsisrelated to the introductionof a speci?c term, called displacement current, that he proposed to add to the set of equations generally assumed to hold at that time, in order to ensure the conservation of the electric charge. The presence of the displacement current permits to describe one of the most - portant phenomenon in electromagnetism, namely, wave propagation; however, in many interesting applications the propagation speed of the wave is very high with respect to the ratio of some typical length and time scale of the considered device, and therefore the dominant aspect becomes the diffusionof the electromagnetic ?elds. When the focus is on diffusioninstead of propagation, from the modelingpointof view this corresponds to neglecting the time derivative of the electric induction (i.e., thedisplacement current introducedby Maxwell)or, alternatively,neglectingthe time derivative of the magnetic induction.

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From the reviews: A comprehensive survey of the magneto-quasistatic reduction of Maxwell's equations-usually called the eddy current approximation. ... There is a comprehensive bibliography of almost 250 items that provides an excellent coverage of the current literature. ... It provides a unique description of some important problems that the authors have studied in recent years and will, I am sure, stimulate a lot more research on related phenomena. The authors are to be commended for their broad perspective and their innovative coverage of these topics. --- (Giles Auchmuty, SIAM Review, Vol. 53 (4), 2011)

From the reviews: A comprehensive survey of the magneto-quasistatic reduction of Maxwell s equations usually called the eddy current approximation. There is a comprehensive bibliography of almost 250 items that provides an excellent coverage of the current literature. It provides a unique description of some important problems that the authors have studied in recent years and will, I am sure, stimulate a lot more research on related phenomena. The authors are to be commended for their broad perspective and their innovative coverage of these topics. (Giles Auchmuty, SIAM Review, Vol. 53 (4), 2011)

Author Information

Since more than ten years the authors have performed researches in the field of numerical approximation of partial differential equations, especially the finite element approximation of problems in electromagnetics. They have published many papers on numerical approximation of partial differential equations on some of the most important journal devoted to this topic. The second author has written two other books on the subject: A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer, 1997 (second edition); A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations, Oxford University Press, 1999

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