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Volume 27, Issue 5
Optimal Error Estimates for Nédélec Edge Elements for Time-Harmonic Maxwell's Equations

Liuqiang Zhong, Shi Shu, Gabriel Wittum & Jinchao Xu

DOI: 10.4208/jcm.2009.27.5.011

J. Comp. Math., 27 (2009), pp. 563-572.

Published online: 2009-10

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  • Abstract

In this paper, we obtain optimal error estimates in both $L^2$-norm and $\boldsymbol{H}$(curl)-norm for the Nédélec 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 $L^2$ error estimates into the $L^2$ 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 Nédélec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

  • Keywords

Edge finite element, Time-harmonic Maxwell's equations.

  • AMS Subject Headings

65N30, 35Q60.

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{JCM-27-563, author = {Liuqiang Zhong, Shi Shu, Gabriel Wittum and Jinchao Xu}, title = {Optimal Error Estimates for Nédélec Edge Elements for Time-Harmonic Maxwell's Equations}, journal = {Journal of Computational Mathematics}, year = {2009}, volume = {27}, number = {5}, pages = {563--572}, abstract = {

In this paper, we obtain optimal error estimates in both $L^2$-norm and $\boldsymbol{H}$(curl)-norm for the Nédélec 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 $L^2$ error estimates into the $L^2$ 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 Nédélec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2009.27.5.011}, url = {http://global-sci.org/intro/article_detail/jcm/8590.html} }
TY - JOUR T1 - Optimal Error Estimates for Nédélec Edge Elements for Time-Harmonic Maxwell's Equations AU - Liuqiang Zhong, Shi Shu, Gabriel Wittum & Jinchao Xu JO - Journal of Computational Mathematics VL - 5 SP - 563 EP - 572 PY - 2009 DA - 2009/10 SN - 27 DO - http://doi.org/10.4208/jcm.2009.27.5.011 UR - https://global-sci.org/intro/article_detail/jcm/8590.html KW - Edge finite element, Time-harmonic Maxwell's equations. AB -

In this paper, we obtain optimal error estimates in both $L^2$-norm and $\boldsymbol{H}$(curl)-norm for the Nédélec 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 $L^2$ error estimates into the $L^2$ 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 Nédélec's second type elements, we present an optimal convergence estimate which improves the best results available in the literature.

Liuqiang Zhong, Shi Shu, Gabriel Wittum and Jinchao Xu. (2009). Optimal Error Estimates for Nédélec Edge Elements for Time-Harmonic Maxwell's Equations. Journal of Computational Mathematics. 27 (5). 563-572. doi:10.4208/jcm.2009.27.5.011
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