This paper proposes a numerical approximation combining finite difference method in time and finite element method in space to solve the two-dimensional time-space fractional Bloch-Torrey equation. Unlike the existing works which focus on the decoupled model, the proposed numerical scheme is established and discussed based on the coupled equations which is from the separation of real and imaginary parts of the original fractional Bloch-Torrey model. In addition, in order to get fast estimation to Caputo fractional derivative and reduce the storage of numerical scheme, an efficient sum-of-exponentials approximation for the kernel $t^{-\alpha },\alpha \in (0,1)$ is adopted. Under the non-uniform time mesh, the stability and convergence are discussed for the semi-discrete scheme, and the error estimate is investigated for the fully discrete scheme in detail. Finally, several numerical tests are provided to verify the correctness of the obtained theoretical results and the effectiveness of our method.

本文提出了一种将时间有限差分法与空间有限元法相结合的数值逼近方法，以求解二维时空分数Bloch-Torrey方程。与现有的专注于解耦模型的工作不同，本提议的数值方案是基于耦合方程建立和讨论的，该耦合方程是从原始分数Bloch-Torrey模型的实部和虚部分离而来的。此外，为了快速估计Caputo分数阶导数并减少数值格式的存储，内核的有效指数和近似${Ť}^{-\alpha }，\alpha \in （0，\mathrm{1个}）$被采用。在非均匀时间网格下，讨论了半离散方案的稳定性和收敛性，并详细研究了全离散方案的误差估计。最后，提供了一些数值测试，以验证所获得的理论结果的正确性和我们方法的有效性。