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Approximation of the eigenvalue problem for the time harmonic Maxwell system by continuous Lagrange finite elements


Authors: Andrea Bonito and Jean-Luc Guermond
Journal: Math. Comp. 80 (2011), 1887-1910
MSC (2010): Primary 65N25, 65F15, 35Q61
DOI: https://doi.org/10.1090/S0025-5718-2011-02464-6
Published electronically: February 4, 2011
MathSciNet review: 2813343
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Abstract: We propose and analyze an approximation technique for the Maxwell eigenvalue problem using $ \mathbf{H}^1$-conforming finite elements. The key idea consists of considering a mixed method controlling the divergence of the electric field in a fractional Sobolev space $ H^{-\alpha}$ with $ \alpha\in (\frac12,1)$. The method is shown to be convergent and spectrally correct.


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Additional Information

Andrea Bonito
Affiliation: Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, Texas 77843
Email: bonito@math.tamu.edu

Jean-Luc Guermond
Affiliation: Department of Mathematics, Texas A&M University, 3368 TAMU, College Station, Texas 77843. On leave from LIMSI, UPR 3251 CNRS, BP 133, 91403 Orsay cedex, France
Email: guermond@math.tamu.edu

DOI: https://doi.org/10.1090/S0025-5718-2011-02464-6
Keywords: Finite elements, Maxwell equations, eigenvalue, spectral approximation.
Received by editor(s): October 1, 2009
Received by editor(s) in revised form: July 12, 2010
Published electronically: February 4, 2011
Additional Notes: The first author was partially supported by the NSF grant DMS-0914977.
The second author was partially supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST)
The third author was partially supported by the NSF grant DMS-07138229
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.