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 - <p style="text-align: justify;">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.</p>