Optimal Error Estimates for Nedelec Edge Elements for Time-Harmonic Maxwell's Equations
Liuqiang Zhong 1, Shi Shu 2, Gabriel Wittum 3, Jinchao Xu 41 School of Mathematical and Computational Sciences, Xiangtan University, Xiangtan 411105, China
2 School of Mathematical and Computational Sciences, Xiangtan University, Xiangtan 411105, China
3 Simulation and Modelling Goethe- Center for Scientic Computing, Goethe-University,Kettenhofweg 139, 60325 Frankfurt am Main, Germany
4 Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA
Received 2008-11-3 Revised 2008-12-12 Accepted 2009-2-5 Online 2009-4-27
In this paper, we obtain optimal error estimates in both L2-norm and H(curl)-norm for the Nedelec edge finite element approximation of the time-harmonic Maxwell's equations on a general Lipschitz domain discretized on quasi-uniform meshes. One key to our proof is to transform the L2 error estimates into the L2 estimate of a discrete divergence-free function which belongs to the edge finite element spaces, and then use the approximation of the discrete divergence-free function by the continuous divergence-free function and a duality argument for the continuous divergence-free function. For Nedelec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.
Key words: Edge finite element, Time-harmonic Maxwell's equations.
AMS subject classifications: 65N30, 35Q60.
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