Volume 37, Issue 4
Unconditional Superconvergence Analysis of an H1-Galerkin Mixed Finite Element Method for Two-Dimensional Ginzburg-Landau Equation

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

Published online: 2019-02

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• Abstract

An H1-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 (Q11 + Q10 × Q01). 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 H1-norm and $\vec{q}$ in H(div; Ω)-norm with order O(h2 + τ2) 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<sup>1</sup>-Galerkin MFEM, Temporal and spatial errors, Unconditionally, Superconvergent results.

65N15, 65N30

dy_shi@zzu.edu.cn (Dongyang Shi)

wjunjun8888@163.com (Junjun Wang)

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@Article{JCM-37-437, author = {Shi , Dongyang and Wang , Junjun }, title = {Unconditional Superconvergence Analysis of an H1-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 H1-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 (Q11 + Q10 × Q01). 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 H1-norm and $\vec{q}$ in H(div; Ω)-norm with order O(h2 + τ2) 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 H1-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://dor.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, H1-Galerkin MFEM, Temporal and spatial errors, Unconditionally, Superconvergent results. AB -

An H1-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 (Q11 + Q10 × Q01). 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 H1-norm and $\vec{q}$ in H(div; Ω)-norm with order O(h2 + τ2) 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.

Dongyang Shi & Junjun Wang. (2019). Unconditional Superconvergence Analysis of an H1-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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