Unconditional Superconvergence Analysis of an $H^1$-Galerkin Mixed Finite Element Method for Two-Dimensional Ginzburg-Landau Equation (Journal of Computational Mathematics, 2019)
An $H^1$-Galerkin mixed finite element method (MFEM) is discussed for the two-dimensionalGinzburg-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 developedand a time-discrete system is introduced to split the error into two parts which are calledthe temporal error and the spatial error, respectively. On one hand, the regularity ofthe time-discrete system is deduced through the temporal error estimation. On the otherhand, 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 proposedmethod. 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.