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Development and Application of Compatible Discretizations of Maxwell’s Equations

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Compatible Spatial Discretizations

Part of the book series: The IMA Volumes in Mathematics and its Applications ((IMA,volume 142))

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. Higher-order 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 semi-orthogonal 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 method-of-lines scheme is used where the Galerkin method reduces the PDE to a semi-discrete 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 Newmark-Beta method, as well as the high order energy conserving explicit Maxwell Symplectic method; for diffusion equations, we employ a generalized Crank-Nicholson method. We conclude with computational examples from resonant cavity problems, time-dependent wave propagation problems, and transient eddy current problems, all obtained using the authors massively parallel computational electromagnetics code EMSolve.

This work was performed under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.

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Author information

Authors and Affiliations

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

    Daniel A. White, Joseph M. Koning & Robert N. Rieben

Authors
  1. Daniel A. White
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  2. Joseph M. Koning
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  3. Robert N. Rieben
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Editor information

Editors and Affiliations

  1. Institute for Mathematics and its Applications, University of Minnesota, Minneapolis, MN, 55455, USA

    Douglas N. Arnold

  2. Computational Mathematics and Algorithms Department, Sandia National Laboratories, Alburquerque, NM, 87185-1110, USA

    Pavel B. Bochev  & Richard B. Lehoucq  & 

  3. Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA, 15213-3890, USA

    Roy A. Nicolaides

  4. Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Mikhail Shashkov

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© 2006 Springer Science+Business Media, LLC

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White, D.A., Koning, J.M., Rieben, R.N. (2006). Development and Application of Compatible Discretizations of Maxwell’s Equations. In: Arnold, D.N., Bochev, P.B., Lehoucq, R.B., Nicolaides, R.A., Shashkov, M. (eds) Compatible Spatial Discretizations. The IMA Volumes in Mathematics and its Applications, vol 142. Springer, New York, NY. https://doi.org/10.1007/0-387-38034-5_11

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