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Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R} ^3$


Authors: P. Monk and L. Demkowicz
Journal: Math. Comp. 70 (2001), 507-523
MSC (2000): Primary 65N30; Secondary 65N15, 65N25
Published electronically: February 23, 2000
MathSciNet review: 1709155
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Abstract:

We analyze the use of edge finite element methods to approximate Maxwell's equations in a bounded cavity. Using the theory of collectively compact operators, we prove $h$-convergence for the source and eigenvalue problems. This is the first proof of convergence of the eigenvalue problem for general edge elements, and it extends and unifies the theory for both problems. The convergence results are based on the discrete compactness property of edge element due to Kikuchi. We extend the original work of Kikuchi by proving that edge elements of all orders possess this property.


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Additional Information

P. Monk
Affiliation: Department of Mathematical Sciences, University of Delaware, Newark DE 19716, USA
Email: monk@math.udel.edu

L. Demkowicz
Affiliation: TICAM, University of Texas at Austin, Austin TX 78712, USA
Email: leszek@brahma.ticam.utexas.edu

DOI: https://doi.org/10.1090/S0025-5718-00-01229-1
Keywords: Finite element methods, discrete compactness, eigenvalues, error estimates
Received by editor(s): October 27, 1998
Received by editor(s) in revised form: April 1, 1999
Published electronically: February 23, 2000
Article copyright: © Copyright 2000 American Mathematical Society