In this paper, we study an effective numerical method for the generalized Ginzburg–Landau equation (GLE). Based on the linearized Crank–Nicolson difference method in time and the standard Galerkin finite element method with bilinear element in space, the time-discrete and space–time discrete systems are both constructed. We focus on a rigorous analysis and consideration of unconditional superconvergence error estimates of the discrete schemes. Firstly, by virtue of the temporal error results, the regularity for the time-discrete system is presented. Secondly, the classical Ritz projection is used to obtain the spatial error with order $O\left(h^2\right)$ in the sense of $L^2$-norm. Thanks to the relationship between the Ritz projection and the interpolated projection, the superclose estimate with order $O({\tau }^2+h^2)$ in the sense of $H^1$-norm is derived. Thirdly, it follows from the interpolated postprocessing technique that the global superconvergence result is deduced. Finally, some numerical results are provided to confirm the theoretical analysis.

在本文中，我们研究了广义Ginzburg-Landau方程（GLE）的有效数值方法。基于时间上的线性Crank–Nicolson差分方法和空间中具有双线性元素的标准Galerkin有限元方法，分别构建了时间离散系统和时空离散系统。我们专注于对离散方案的无条件超收敛误差估计进行严格的分析和考虑。首先，根据时间误差结果，给出了时间离散系统的规律性。其次，使用经典的Ritz投影获得阶次的空间误差$Ø（H^2）$ 在某种意义上 ${\mathrm{大号}}^2$-规范。得益于Ritz投影和插值投影之间的关系，带阶数的超闭合估计$Ø（{\tau }^2+H^2）$ 在某种意义上 $H^{\mathrm{1个}}$-范数派生。第三，从插值后处理技术推论得出了全局超收敛结果。最后，提供一些数值结果以证实理论分析。