An unstructured mesh finite difference/finite element method for the three-dimensional time-space fractional Bloch-Torrey equations on irregular domains

In this paper, we use the finite element method (FEM) to solve the time-space fractional Bloch-Torrey equation on irregular domains in R³. Based on linear Lagrange basis functions, a space semi-discrete FEM scheme is given. By adopting the L2−1_σ approximation for the Caputo fractional derivative, a fully discrete scheme is presented. Furthermore, we provide the details on how to implement our FEM for the space fractional Bloch-Torrey equation. Also, the stability and convergence of the fully discrete scheme is investigated. The error estimations with respect to the L² and energy norms are given. In addition, some numerical examples are presented to verify the efficiency of our method.