Numerical Analysis of Electromagnetic Problems
We give theoretical and computational overview of numerical analysis of the finite element methods for electromagnetics. In particular, theoretical comments on the edge and face elements, frequently employed in the finite element discretizations, are give
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1
Introduction
Numerical analysis of electromagnetic problems is now quite important in wide fields of science and engineering. The application of the finite element method (FEM) to such ends is very effective especially for 3-D problems since FEM is well suited for complex regions with various boundary conditions. By appropriate modeling of the Maxwell equations of electromagnetics, we have a variety of problems describing electromagnetic phenomena in practice. In FEM, we first derive weak forms to such problems, and then obtain discrete equations by the finite element procedures. In this process, we often utilize mixed formulations based on the Lagrange multiplier techniques. Then the Nedelec type edge elements and the Raviart-Thomas type face ones are very effective to approximate vector fields with their rotations and/or divergences . They are also well suited to the tangential and normal boundary conditions in electromagnetics, and their effectiveness is now widely recognized through practical experiences. Although there have been proposed various approaches without relying on such vector elements, most of them have been unsuccessful as is known the "nightmare" in computational electromagnetics. The Raviart-Thomas face elements were proposed in [26] , and they have been generalized in various fashions as summarized in [7]. On the other hand , basic edge elements of simplex or cube-based shapes were early proposed by Nedelec [23],124], and then their generalizations such as the covariant interpolation elements have been developed [6], [27] . Once the discrete equations are obtained, they must be solved by means of appropriate computational methods. Since the equations are usually of very large-scale especially in 3-D cases, we often prefer to reliable iterative methods even in linear cases. The outline of this note is as follows. We first present some basic electromagnetic problems and give them variational formulations , most of them I. Babuška et al. (eds.), Mathematical Modeling and Numerical Simulation in Continuum Mechanics © Springer-Verlag Berlin Heidelberg 2002
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Fumio Kikuchi
are of mixed type, either explicitly or implicitly, and easy to implement as the finite element methods. Then we show the outline of theoretical numerical analysis of such finite element schemes. Among them, we give comments on the discrete compactness properties, which are discrete analogs of the compactness properties of electromagnetic spaces and difficult to check theoretically [13] . Finally, we present some computational techniques to solve discrete equations arising from the finite element discretizations.
2
Electromagnetic problems
First we will explain typical electromagnetic problems with some variational formulations. 2.1
Maxwell's equations
As is well known, the governing equations of electromagnetics are the Maxwell partial differential equations, which may be expressed as follows when considered in a 3-D domain n occupied by the medium :
aB H aD . rotE+7jt=O, divD=p, rot - at =3 , divB=O in n,
(1)
whe
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