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Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation

Unconditionally Superclose Analysis of a New Mixed Finite Element Method for Nonlinear Parabolic Equation
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Year:    2019

Author:    Dongyang Shi, Fengna Yan, Junjun Wang

Journal of Computational Mathematics, Vol. 37 (2019), Iss. 1 : pp. 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.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/jcm.1707-m2016-0718

Journal of Computational Mathematics, Vol. 37 (2019), Iss. 1 : pp. 1–17

Published online:    2019-01

AMS Subject Headings:   

Copyright:    COPYRIGHT: © Global Science Press

Pages:    17

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

Author Details

Dongyang Shi

Fengna Yan

Junjun Wang

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