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In this paper, a linearized Crank-Nicolson Galerkin finite element method(FEM) for generalized Ginzburg-Landau equation (GLE) is considered, in which,the difference method in time and the standard Galerkin FEM are employed. Basedon the linearized Crank-Nicolson difference method in time and the standardGalerkin finite element method with bilinear element in space, thetime-discrete and space-time discrete systems are both constructed. We focus ona rigorous analysis and consideration of unconditional superconvergence errorestimates of the discrete schemes. Firstly, by virtue of the temporal errorresults, the regularity for the time-discrete system is presented. Secondly,the classical Ritz projection is used to obtain the spatial error with order$O(h^2)$ in the sense of $L^2-$norm. Thanks to the relationship between theRitz projection and the interpolated projection, the superclose estimate withorder $O(\tau^2 + h^2)$ in the sense of $H^1-$norm is derived. Thirdly, itfollows from the interpolated postprocessing technique that the globalsuperconvergence result is deduced. Finally, some numerical results areprovided to confirm the theoretical analysis.

Unconditional superconvergence analysis of a linearized Crank-Nicolson Galerkin FEM for generalized Ginzburg-Landau equation