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Mathematics of Computation

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A posteriori error estimates for Maxwell equations

Author: Joachim Schöberl
Journal: Math. Comp. 77 (2008), 633-649
MSC (2000): Primary 65N30
Published electronically: December 12, 2007
MathSciNet review: 2373173
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Abstract: Maxwell equations are posed as variational boundary value problems in the function space $ H(\operatorname{curl})$ and are discretized by Nédélec finite elements. In Beck et al., 2000, a residual type a posteriori error estimator was proposed and analyzed under certain conditions onto the domain. In the present paper, we prove the reliability of that error estimator on Lipschitz domains. The key is to establish new error estimates for the commuting quasi-interpolation operators recently introduced in J. Schöberl, Commuting quasi-interpolation operators for mixed finite elements. Similar estimates are required for additive Schwarz preconditioning. To incorporate boundary conditions, we establish a new extension result.

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Additional Information

Joachim Schöberl
Affiliation: Center for Computational Engineering Science, RWTH Aachen University, Pauwelstrasse 19, D-52074 Aachen, Germany

Keywords: Cl\'ement operator, Maxwell equations, edge elements
Received by editor(s): May 5, 2005
Received by editor(s) in revised form: July 25, 2006
Published electronically: December 12, 2007
Additional Notes: The author acknowledges support from the Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austria, and from the Austrian Science Foundation FWF within project grant Start Y-192, “hp-FEM: Fast Solvers and Adaptivity”
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.