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

Abstract

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 L2−1σ 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.