In this paper, we propose a novel numerical technique to 2D multi-term time and space fractional Bloch–Torrey equations defined on an irregular convex domain. First, we consider the problem with space integral Laplacian operator. We present the semi-discrete and fully-discrete schemes by using the $L1$ formula on a temporal graded mesh and an unstructured-mesh Galerkin finite element method (FEM) based on the matrix transfer technique (MTT). Moreover, the unconditional stability and convergence of the schemes are discussed and theoretically proved. Then we extend the technique and develop a numerical scheme for the case of space fractional Laplacian operator. In addition, the recent $L{1}^{+}$ formula is utilized to improving the temporal accuracy. Finally, three numerical examples in rectangular, elliptical and human brain-like irregular domains are given to demonstrate the accuracy and efficiency of the proposed numerical scheme.

在本文中，我们提出了一种新的数值技术，用于对不规则凸域上定义的二维多维时空分数Bloch-Torrey方程。首先，我们考虑空间积分拉普拉斯算子的问题。我们通过使用$\mathrm{大号}\mathrm{1个}$基于矩阵传递技术（MTT）的时间渐变网格和非结构化网格Galerkin有限元方法（FEM）的公式。此外，对方案的无条件稳定性和收敛性进行了讨论和理论证明。然后，我们扩展了该技术，并针对空间分数拉普拉斯算子的情况开发了一种数值方案。另外，最近$\mathrm{大号}{\mathrm{1个}}^{+}$公式用于提高时间精度。最后，给出了矩形，椭圆形和人脑状不规则域中的三个数值示例，以证明所提出的数值方案的准确性和效率。