TT-M finite element algorithm for a two-dimensional space fractional Gray–Scott model

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.