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The approximation of the Maxwell eigenvalue problem using a least-squares method


Authors: James H. Bramble, Tzanio V. Kolev and Joseph E. Pasciak
Journal: Math. Comp. 74 (2005), 1575-1598
MSC (2000): Primary 65F10, 65N30
DOI: https://doi.org/10.1090/S0025-5718-05-01759-X
Published electronically: May 5, 2005
MathSciNet review: 2164087
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Abstract: In this paper we consider an approximation to the Maxwell's eigenvalue problem based on a very weak formulation of two div-curl systems with complementary boundary conditions. We formulate each of these div-curl systems as a general variational problem with different test and trial spaces, i.e., the solution space is ${\boldsymbol {L}^2(\Omega)}\equiv (L^2(\Omega))^3$ and components in the test spaces are in subspaces of $H^1(\Omega)$, the Sobolev space of order one on the computational domain $\Omega$. A finite-element least-squares approximation to these variational problems is used as a basis for the approximation. Using the structure of the continuous eigenvalue problem, a discrete approximation to the eigenvalues is set up involving only the approximation to either of the div-curl systems. We give some theorems that guarantee the convergence of the eigenvalues to those of the continuous problem without the occurrence of spurious values. Finally, some results of numerical experiments are given.


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Additional Information

James H. Bramble
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: bramble@math.tamu.edu

Tzanio V. Kolev
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: tkolev@math.tamu.edu

Joseph E. Pasciak
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843-3368
Email: pasciak@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-05-01759-X
Keywords: Maxwell eigenvalues, div-curl systems, inf-sup condition, finite element approximation, negative norm least-squares, Maxwell's equations
Received by editor(s): April 23, 2004
Received by editor(s) in revised form: October 12, 2004
Published electronically: May 5, 2005
Additional Notes: This work is based upon work supported by the National Science Foundation under grant No. 0311902.
Article copyright: © Copyright 2005 American Mathematical Society

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