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arXiv · 1806.09157

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

Meng Li·Dongyang Shi·Junjun Wang
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Abstract

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. 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 $O(h^2)$ 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.

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Meng Li, Dongyang Shi, Junjun Wang. 2018-06-24. Unconditional superconvergence analysis of a linearized Crank-Nicolson Galerkin FEM for generalized Ginzburg-Landau equation. https://arxiv.org/abs/1806.09157

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