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 the difference scheme is uniquely solvable, stable and convergent under mild conditions. The optimal convergence order O(τ 2 + h2x + h2y ) is obtained in the pointwise sense by developing a new two-dimensional fractional Sobolev imbedding inequality based on the work in [K. Kirkpatrick, E. Lenzmann, G. Staffilani, Commun. Math. Phys., 317 (2013), pp. 563–591], 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.