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Volume 40, Issue 3
A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem

Xiaonian Long & Qianqian Ding

DOI: 10.4208/jcm.2010-m2020-0145

J. Comp. Math., 40 (2022), pp. 354-372.

Published online: 2022-02

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  • Abstract

In this paper, we study the finite element approximation for nonlinear thermal equation. Because the nonlinearity of the equation, our theoretical analysis is based on the error of temporal and spatial discretization. We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential, and establish optimal $L^2$ error estimates for the fully discrete finite element solution without any restriction on the time-step size. The discrete solution is bounded in infinite norm. Finally, several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

  • Keywords

Thermal equation, Joule heating, Finite element method, Unconditional convergence, Second order backward difference formula, Optimal $L^2$-estimate.

  • AMS Subject Headings

65M12, 65M15, 65M60, 35K61

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

longxiaonian@lsec.cc.ac.cn (Xiaonian Long)

dingqianqian@lsec.cc.ac.cn (Qianqian Ding)

  • BibTex
  • RIS
  • TXT
@Article{JCM-40-354, author = {Long , Xiaonian and Ding , Qianqian}, title = {A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem}, journal = {Journal of Computational Mathematics}, year = {2022}, volume = {40}, number = {3}, pages = {354--372}, abstract = {

In this paper, we study the finite element approximation for nonlinear thermal equation. Because the nonlinearity of the equation, our theoretical analysis is based on the error of temporal and spatial discretization. We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential, and establish optimal $L^2$ error estimates for the fully discrete finite element solution without any restriction on the time-step size. The discrete solution is bounded in infinite norm. Finally, several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.2010-m2020-0145}, url = {http://global-sci.org/intro/article_detail/jcm/20241.html} }
TY - JOUR T1 - A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem AU - Long , Xiaonian AU - Ding , Qianqian JO - Journal of Computational Mathematics VL - 3 SP - 354 EP - 372 PY - 2022 DA - 2022/02 SN - 40 DO - http://doi.org/10.4208/jcm.2010-m2020-0145 UR - https://global-sci.org/intro/article_detail/jcm/20241.html KW - Thermal equation, Joule heating, Finite element method, Unconditional convergence, Second order backward difference formula, Optimal $L^2$-estimate. AB -

In this paper, we study the finite element approximation for nonlinear thermal equation. Because the nonlinearity of the equation, our theoretical analysis is based on the error of temporal and spatial discretization. We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential, and establish optimal $L^2$ error estimates for the fully discrete finite element solution without any restriction on the time-step size. The discrete solution is bounded in infinite norm. Finally, several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

Long , Xiaonian and Ding , Qianqian. (2022). A Second Order Unconditionally Convergent Finite Element Method for the Thermal Equation with Joule Heating Problem. Journal of Computational Mathematics. 40 (3). 354-372. doi:10.4208/jcm.2010-m2020-0145
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