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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 = {}, 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 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 & Jinchao Xu. (2019). 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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