Research paperA 2D multi-term time and space fractional Bloch-Torrey model based on bilinear rectangular finite elements

•A 2D multi-term time and space fractional Bloch-Torrey equation (2D-MTTS-FBTE) is studied•Galerkin finite element method for the 2D-MTTS-FBTE is proposed.•A rigorous analysis of stability and error estimation is provided.•The scheme is applied to solve the multi-term time and space fractional Bloch-Torrey equation.•The proposed discrete scheme can be extended to the case of unstructured quadrilateral meshes by choosing the basis functions accordingly.

The consideration of diffusion processes in magnetic resonance imaging (MRI) signal attenuation is classically described by the Bloch-Torrey equation. However, many recent works highlight the distinct deviation in MRI signal decay due to anomalous diffusion, which motivates the fractional order generalization of the Bloch-Torrey equation.In this work, we study the two-dimensional multi-term time and space fractional diffusion equation generalized from the time and space fractional Bloch-Torrey equation. By using the Galerkin finite element method with a structured mesh consisting of rectangular elements to discretize in space and the L1 approximation of the Caputo fractional derivative in time, a fully discrete numerical scheme is derived. A rigorous analysis of stability and error estimation is provided. Numerical experiments in the square and L-shaped domains are performed to give an insight into the efficiency and reliability of our method. Then the scheme is applied to solve the multi-term time and space fractional Bloch-Torrey equation, which shows that the extra time derivative terms impact the relaxation process.