Development and Application of Compatible Discretizations of Maxwell’s Equations
 Daniel A. White,
 Joseph M. Koning,
 Robert N. Rieben
 … show all 3 hide
Abstract
We present the development and application of compatible finite element discretizations of electromagnetics problems derived from the time dependent, full wave Maxwell equations. We review the H(curl)conforming finite element method, using the concepts and notations of differential forms as a theoretical framework. We chose this approach because it can handle complex geometries, it is free of spurious modes, it is numerically stable without the need for filtering or artificial diffusion, it correctly models the discontinuity of fields across material boundaries, and it can be very high order. Higherorder H(curl) and H(div) conforming basis functions are not unique and we have designed an extensible C++ framework that supports a variety of specific instantiations of these such as standard interpolatory bases, spectral bases, hierarchical bases, and semiorthogonal bases. Virtually any electromagnetics problem that can be cast in the language of differential forms can be solved using our framework. For time dependent problems a methodoflines scheme is used where the Galerkin method reduces the PDE to a semidiscrete system of ODE’s, which are then integrated in time using finite difference methods. For time integration of wave equations we employ the unconditionally stable implicit NewmarkBeta method, as well as the high order energy conserving explicit Maxwell Symplectic method; for diffusion equations, we employ a generalized CrankNicholson method. We conclude with computational examples from resonant cavity problems, timedependent wave propagation problems, and transient eddy current problems, all obtained using the authors massively parallel computational electromagnetics code EMSolve.
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 Title
 Development and Application of Compatible Discretizations of Maxwell’s Equations
 Book Title
 Compatible Spatial Discretizations
 Pages
 pp 209234
 Copyright
 2006
 DOI
 10.1007/0387380345_11
 Print ISBN
 9780387309163
 Online ISBN
 9780387380346
 Series Title
 The IMA Volumes in Mathematics and its Applications
 Series Volume
 142
 Series ISSN
 09406573
 Publisher
 Springer New York
 Copyright Holder
 Springer Science+Business Media, LLC
 Additional Links
 Topics
 Keywords

 Computational electromagnetics
 Maxwell’s equations
 vector finite elements
 high order methods
 H(curl) and H(div)  conforming methods
 discrete differential forms
 spurious modes
 numerical dispersion
 wave propagation
 transient eddy currents
 electromagnetic diffusion
 eBook Packages
 Editors

 Douglas N. Arnold ^{(2)}
 Pavel B. Bochev ^{(3)}
 Richard B. Lehoucq ^{(3)}
 Roy A. Nicolaides ^{(4)}
 Mikhail Shashkov ^{(5)}
 Editor Affiliations

 2. Institute for Mathematics and its Applications, University of Minnesota
 3. Computational Mathematics and Algorithms Department, Sandia National Laboratories
 4. Department of Mathematical Sciences, Carnegie Mellon University
 5. Theoretical Division, Los Alamos National Laboratory
 Authors

 Daniel A. White ^{(6)}
 Joseph M. Koning ^{(6)}
 Robert N. Rieben ^{(6)}
 Author Affiliations

 6. Defense Sciences Engineering Division, Lawrence Livermore National Laboratory, Livermore, CA, 94551
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