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

Abstract

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