Galerkin finite element method for two-dimensional space and time fractional Bloch-Torrey equation

In this paper, we consider the Galerkin finite element method for the two-dimensional space and time fractional Bloch-Torrey equation (2D-STFBTE). Utilizing the L2−1σ formula to discretize temporal Caputo derivative, we obtain the local truncation error with O(τ3−α) in the temporal direction, where α is the order of time fractional derivative. Furthermore, the semi-discrete form for the problem is given and, the stability and convergence of the semi-discrete variational formulation are rigorously proved in L2 norm and fractional norm respectively. Then, we derive a fully discrete scheme for the 2D-STFBTE and investigate its convergence theoretically. Finally, extensive numerical results based on linear piecewise polynomials verify the theoretical results and show the effectiveness of the proposed scheme.