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 L2-1 sigma formula to discretize temporal Caputo derivative, we obtain the local truncation error with O (iota H3-Q) in the temporal direction, where a 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 L-2 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. (C) 2017 Elsevier Inc. All rights reserved.