Finite difference/finite element method for two-dimensional time-space fractional Bloch-Torrey equations with variable coefficients on irregular convex domains

In magnetic resonance imaging of the human brain, the diffusion process of tissue water is considered in the complex tissue environment of cells, membranes and connective tissue. Models based on fractional order Bloch-Torrey equations are known to provide insights into tissue structures and the microenvironment. In this paper, we consider new two-dimensional multi-term time and space fractional Bloch-Torrey equations with variable coefficients on irregular convex domains, which involve the Caputo time fractional derivative and the Riemann-Liouville space fractional derivative. An unstructured-mesh Galerkin finite element method is used to discretize the spatial fractional derivative, while for each time fractional derivative we use the L1 scheme on a temporal graded mesh. The stability and convergence of the fully discrete scheme are proved. Numerical examples are given to verify the efficiency of our method. (C) 2020 Elsevier Ltd. All rights reserved.