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Volume 37, Issue 1
Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation

Dongyang Shi, Fengna Yan & Junjun Wang

DOI: 10.4208/jcm.1707-m2016-0718

J. Comp. Math., 37 (2019), pp. 1-17.

Published online: 2018-08

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  • Abstract

This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example, a new mixed finite element method (FEM) is established and the $τ$ -independent superclose results of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{\rightarrow}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is an error splitting technique, with which the time-discrete and the spatial-discrete systems are constructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or require certain time step conditions. Finally, some numerical results are provided to confirm the theoretical analysis, and show the efficiency of the proposed method.

  • Keywords

Nonlinear parabolic equation, Mixed FEM, Time-discrete and spatial-discrete systems, $τ$-independent superclose results.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

dy_shi@zzu.edu.cn (Dongyang Shi)

yanfengnaa@163.com (Fengna Yan)

wjunjun8888@163.com (Junjun Wang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-37-1, author = {Shi , DongyangYan , Fengna and Wang , Junjun}, title = {Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation}, journal = {Journal of Computational Mathematics}, year = {2018}, volume = {37}, number = {1}, pages = {1--17}, abstract = {

This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example, a new mixed finite element method (FEM) is established and the $τ$ -independent superclose results of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{\rightarrow}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is an error splitting technique, with which the time-discrete and the spatial-discrete systems are constructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or require certain time step conditions. Finally, some numerical results are provided to confirm the theoretical analysis, and show the efficiency of the proposed method.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1707-m2016-0718}, url = {http://global-sci.org/intro/article_detail/jcm/12645.html} }
TY - JOUR T1 - Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation AU - Shi , Dongyang AU - Yan , Fengna AU - Wang , Junjun JO - Journal of Computational Mathematics VL - 1 SP - 1 EP - 17 PY - 2018 DA - 2018/08 SN - 37 DO - http://doi.org/10.4208/jcm.1707-m2016-0718 UR - https://global-sci.org/intro/article_detail/jcm/12645.html KW - Nonlinear parabolic equation, Mixed FEM, Time-discrete and spatial-discrete systems, $τ$-independent superclose results. AB -

This paper develops a framework to deal with the unconditional superclose analysis of nonlinear parabolic equation. Taking the finite element pair $Q_{11}/Q_{01} × Q_{10}$ as an example, a new mixed finite element method (FEM) is established and the $τ$ -independent superclose results of the original variable $u$ in $H^1$-norm and the flux variable $\mathop{q} \limits ^{\rightarrow}= −a(u)∇u$ in $L^2$-norm are deduced ($τ$ is the temporal partition parameter). A key to our analysis is an error splitting technique, with which the time-discrete and the spatial-discrete systems are constructed, respectively. For the first system, the boundedness of the temporal errors is obtained. For the second system, the spatial superclose results are presented unconditionally, while the previous literature always only obtain the convergent estimates or require certain time step conditions. Finally, some numerical results are provided to confirm the theoretical analysis, and show the efficiency of the proposed method.

Shi , DongyangYan , Fengna and Wang , Junjun. (2018). Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation. Journal of Computational Mathematics. 37 (1). 1-17. doi:10.4208/jcm.1707-m2016-0718
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