In this article, a fast time two-mesh (TT-M) finite element (FE) method for the two-dimensional space fractional Gray–Scott model is studied and discussed to get the numerical solutions effectively. The method mainly includes three steps: firstly, one uses an iterative method for solving the coupled nonlinear system on the time coarse grid; secondly, by an interpolation formula, one can get any coarse values on the time fine mesh; finally, based on the computed coarser solutions, a linear FE system on time fine mesh can be constructed by using the two-variables Taylor’s formula. Here, some theoretical results, which include stability and a priori error for the fully discrete scheme, are analyzed and proved. Furthermore, the computing data are given to verify the correctness of the theoretical results and to illustrate that the TT-M FE algorithm can reduce the computing time.

在本文中，研究和讨论了二维空间分数式Gray-Scott模型的快速两网格（TT-M）有限元（FE）方法，以有效地获得数值解。该方法主要包括三个步骤：首先，使用迭代方法求解时间粗网格上的耦合非线性系统。其次，通过插值公式，可以在时间精细网格上获得任何粗略值。最后，基于计算出的粗略解，可以使用二变量泰勒公式构造基于时间精细网格的线性有限元系统。在此，分析并证明了一些理论结果，包括稳定性和完全离散方案的先验误差。此外，