Unconditional Superconvergence Analysis of an $H^1$-Galerkin Mixed Finite Element Method for Two-Dimensional Ginzburg-Landau Equation
Year: 2019
Author: Dongyang Shi, Junjun Wang
Journal of Computational Mathematics, Vol. 37 (2019), Iss. 4 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jcm.1802-m2017-0198
Journal of Computational Mathematics, Vol. 37 (2019), Iss. 4 : pp. 437–457
Published online: 2019-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 21
Keywords: The two-dimensional Ginzburg-Landau equation $H^1$-Galerkin MFEM Temporal and spatial errors Unconditionally Superconvergent results.