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Volume 12, Issue 1
Superconvergence Analysis for the Maxwell's Equations in Debye Medium with a Thermal Effect

Changhui Yao, Dongyang Shi & Mengmeng Hou

Adv. Appl. Math. Mech., 12 (2020), pp. 141-163.

Published online: 2019-12

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  • Abstract

In this paper, a mixed finite element method is investigated for the Maxwell's equations in Debye medium with a thermal effect. In particular, in two dimensional case, the zero order Nédélec element $(Q_{01}\times Q_{10})$, the piecewise constant space $Q_0$ element, and the bilinear element $Q_{11}$ are used to approximate the electric field E and the polarization electric field P, the magnetic field H, and the temperature field $u$, respectively. With the help of the high accuracy results, mean-value technique and interpolation postprocessing approach, the convergent rate $\mathcal{O}(\tau+h^2)$ for global superconvergence results are obtained under the time step constraint $\tau=\mathcal{O}(h^{1+\gamma}),$ $ \gamma>0$ by using the linearized backward $Euler$ finite element discrete scheme. At last, a numerical experiment is given to verify the theoretical analysis and the validity of our method.

  • AMS Subject Headings

65N30, 65N15

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

chyao@lsec.cc.ac.cn (Changhui Yao)

dy_shi@zzu.edu.cn (Dongyang Shi)

2605822173@qq.com (Mengmeng Hou)

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@Article{AAMM-12-141, author = {Yao , ChanghuiShi , Dongyang and Hou , Mengmeng}, title = {Superconvergence Analysis for the Maxwell's Equations in Debye Medium with a Thermal Effect}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2019}, volume = {12}, number = {1}, pages = {141--163}, abstract = {

In this paper, a mixed finite element method is investigated for the Maxwell's equations in Debye medium with a thermal effect. In particular, in two dimensional case, the zero order Nédélec element $(Q_{01}\times Q_{10})$, the piecewise constant space $Q_0$ element, and the bilinear element $Q_{11}$ are used to approximate the electric field E and the polarization electric field P, the magnetic field H, and the temperature field $u$, respectively. With the help of the high accuracy results, mean-value technique and interpolation postprocessing approach, the convergent rate $\mathcal{O}(\tau+h^2)$ for global superconvergence results are obtained under the time step constraint $\tau=\mathcal{O}(h^{1+\gamma}),$ $ \gamma>0$ by using the linearized backward $Euler$ finite element discrete scheme. At last, a numerical experiment is given to verify the theoretical analysis and the validity of our method.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2019-0126}, url = {http://global-sci.org/intro/article_detail/aamm/13422.html} }
TY - JOUR T1 - Superconvergence Analysis for the Maxwell's Equations in Debye Medium with a Thermal Effect AU - Yao , Changhui AU - Shi , Dongyang AU - Hou , Mengmeng JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 141 EP - 163 PY - 2019 DA - 2019/12 SN - 12 DO - http://doi.org/10.4208/aamm.OA-2019-0126 UR - https://global-sci.org/intro/article_detail/aamm/13422.html KW - Maxwell's equations, thermal effect, error analysis, superconvergence. AB -

In this paper, a mixed finite element method is investigated for the Maxwell's equations in Debye medium with a thermal effect. In particular, in two dimensional case, the zero order Nédélec element $(Q_{01}\times Q_{10})$, the piecewise constant space $Q_0$ element, and the bilinear element $Q_{11}$ are used to approximate the electric field E and the polarization electric field P, the magnetic field H, and the temperature field $u$, respectively. With the help of the high accuracy results, mean-value technique and interpolation postprocessing approach, the convergent rate $\mathcal{O}(\tau+h^2)$ for global superconvergence results are obtained under the time step constraint $\tau=\mathcal{O}(h^{1+\gamma}),$ $ \gamma>0$ by using the linearized backward $Euler$ finite element discrete scheme. At last, a numerical experiment is given to verify the theoretical analysis and the validity of our method.

Changhui Yao, Dongyang Shi & Mengmeng Hou. (2019). Superconvergence Analysis for the Maxwell's Equations in Debye Medium with a Thermal Effect. Advances in Applied Mathematics and Mechanics. 12 (1). 141-163. doi:10.4208/aamm.OA-2019-0126
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