Pointwise error estimate in difference setting for the two-dimensional nonlinear fractional complex Ginzburg-Landau equation

In this paper, we propose a three-level linearized implicit difference scheme for the two-dimensional spatial fractional nonlinear complex Ginzburg-Landau equation. We prove that is stable and convergent under mild conditions. The optimal convergence order $\mathcal {O}(\tau ^{2}+{h_{x}^{2}}+{h_{y}^{2}})$ obtained in pointwise sense by developing new Sobolev imbedding inequality based on work Kirkpatrick et al. (Commun. Math. Phys. 317, 563–591 2013), an energy argument careful attention to term. Numerical examples are presented verify validity of theoretical results different choices orders ? ?.