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Volume 37, Issue 4
Unconditional Superconvergence Analysis of an $H^1$-Galerkin Mixed Finite Element Method for Two-Dimensional Ginzburg-Landau Equation

Dongyang Shi & Junjun Wang

DOI: 10.4208/jcm.1802-m2017-0198

J. Comp. Math., 37 (2019), pp. 437-457.

Published online: 2019-02

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  • Abstract

An $H^1$-Galerkin mixed finite element method (MFEM) is discussed for the two-dimensional Ginzburg-Landau equation with the bilinear element and zero order Raviart-Thomas element $(Q_{11} + Q_{10} × Q_{01})$. A linearized Crank-Nicolson fully-discrete scheme is developed and a time-discrete system is introduced to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, the regularity of the time-discrete system is deduced through the temporal error estimation. On the other hand, the superconvergent estimates of $u$ in $H^1$-norm and $\vec{q}$ in $H$(div; Ω)-norm with order $O(h^2 + τ^2)$ are obtained unconditionally based on the achievement of the spatial result. At last, a numerical experiment is included to illustrate the feasibility of the proposed method. Here, $h$ is the subdivision parameter and $τ$ is the time step.

  • Keywords

The two-dimensional Ginzburg-Landau equation, $H^1$-Galerkin MFEM, Temporal and spatial errors, Unconditionally, Superconvergent results.

  • AMS Subject Headings

65N15, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

  • Email address

dy_shi@zzu.edu.cn (Dongyang Shi)

wjunjun8888@163.com (Junjun Wang)

  • BibTex
  • RIS
  • TXT
@Article{JCM-37-437, author = {Shi , Dongyang and Wang , Junjun}, title = {Unconditional Superconvergence Analysis of an $H^1$-Galerkin Mixed Finite Element Method for Two-Dimensional Ginzburg-Landau Equation}, journal = {Journal of Computational Mathematics}, year = {2019}, volume = {37}, number = {4}, pages = {437--457}, abstract = {

An $H^1$-Galerkin mixed finite element method (MFEM) is discussed for the two-dimensional Ginzburg-Landau equation with the bilinear element and zero order Raviart-Thomas element $(Q_{11} + Q_{10} × Q_{01})$. A linearized Crank-Nicolson fully-discrete scheme is developed and a time-discrete system is introduced to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, the regularity of the time-discrete system is deduced through the temporal error estimation. On the other hand, the superconvergent estimates of $u$ in $H^1$-norm and $\vec{q}$ in $H$(div; Ω)-norm with order $O(h^2 + τ^2)$ are obtained unconditionally based on the achievement of the spatial result. At last, a numerical experiment is included to illustrate the feasibility of the proposed method. Here, $h$ is the subdivision parameter and $τ$ is the time step.

}, issn = {1991-7139}, doi = {https://doi.org/10.4208/jcm.1802-m2017-0198}, url = {http://global-sci.org/intro/article_detail/jcm/13001.html} }
TY - JOUR T1 - Unconditional Superconvergence Analysis of an $H^1$-Galerkin Mixed Finite Element Method for Two-Dimensional Ginzburg-Landau Equation AU - Shi , Dongyang AU - Wang , Junjun JO - Journal of Computational Mathematics VL - 4 SP - 437 EP - 457 PY - 2019 DA - 2019/02 SN - 37 DO - http://doi.org/10.4208/jcm.1802-m2017-0198 UR - https://global-sci.org/intro/article_detail/jcm/13001.html KW - The two-dimensional Ginzburg-Landau equation, $H^1$-Galerkin MFEM, Temporal and spatial errors, Unconditionally, Superconvergent results. AB -

An $H^1$-Galerkin mixed finite element method (MFEM) is discussed for the two-dimensional Ginzburg-Landau equation with the bilinear element and zero order Raviart-Thomas element $(Q_{11} + Q_{10} × Q_{01})$. A linearized Crank-Nicolson fully-discrete scheme is developed and a time-discrete system is introduced to split the error into two parts which are called the temporal error and the spatial error, respectively. On one hand, the regularity of the time-discrete system is deduced through the temporal error estimation. On the other hand, the superconvergent estimates of $u$ in $H^1$-norm and $\vec{q}$ in $H$(div; Ω)-norm with order $O(h^2 + τ^2)$ are obtained unconditionally based on the achievement of the spatial result. At last, a numerical experiment is included to illustrate the feasibility of the proposed method. Here, $h$ is the subdivision parameter and $τ$ is the time step.

Shi , Dongyang and Wang , Junjun. (2019). Unconditional Superconvergence Analysis of an $H^1$-Galerkin Mixed Finite Element Method for Two-Dimensional Ginzburg-Landau Equation. Journal of Computational Mathematics. 37 (4). 437-457. doi:10.4208/jcm.1802-m2017-0198
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