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.

在本文中，我们为二维空间分数非线性复Ginzburg-Landau方程提出了一个三级线性化隐式差分格式。我们证明了差分方案在温和条件下是稳定且收敛的。最佳收敛阶数\（\ mathcal {O}（\ tau ^ {2} + {h_ {x} ^ {2}} + {h_ {y} ^ {2}}）\）在逐点意义上通过根据Kirkpatrick等人的工作，开发了一个新的二维分数阶Sobolev嵌入不等式。（COMMUN。数学物理学，317，563-591 2013），能量参数和非线性项小心注意。通过数值例子验证了不同分数阶α和β选择的理论结果的有效性。