In this paper, an unconditionally stable fully discrete numerical scheme for the two-dimensional (2D) time fractional variable coefficient diffusion equations with non-smooth solutions is constructed and analyzed. The $L2$-$1_{\sigma }$ scheme is applied for the discretization of time fractional derivative on graded meshes and anisotropic finite element method (FEM) is employed for the spatial discretization. The unconditional stability and convergence of the proposed scheme are proved rigorously. It is shown that the order $O(h^2+N^{-min\{r\alpha ,2\}})$ can be achieved, where $h$ is the spatial step, $N$ is the number of partition in temporal direction, $r$ is the temporal meshes grading parameter and $\alpha$ is the order of fractional derivative. A numerical example is provided to verify the sharpness of our error analysis.

本文针对具有非光滑解的二维（2D）时间分数可变系数扩散方程，建立了无条件稳定的全离散数值格式。的$\mathrm{大号}2$--${\mathrm{1个}}_{\sigma }$该方案被用于时间分数阶导数在梯度网格上的离散化，而各向异性有限元方法（FEM）被用于空间离散化。严格证明了所提方案的无条件稳定性和收敛性。显示顺序$Ø（H^2+{ñ}^{-分\{\mathrm{[R}\alpha ，2\}}）$ 可以实现的地方 $H$ 是空间的一步 $ñ$ 是时间方向上的分区数， $\mathrm{[R}$ 是时间网格分级参数，并且 $\alpha$是分数导数的阶数。提供了一个数字示例来验证我们的误差分析的准确性。