Finite difference/finite element method for two-dimensional space and time fractional Bloch-Torrey equations

In this paper, a class of two-dimensional space and time fractional Bloch-Torrey equations (2D-STFBTEs) are considered. Some definitions and properties of fractional derivative spaces are presented. By finite difference method and Galerkin finite element method, a semi-discrete variational formulation for 2D-STFBTEs is obtained. The stability and convergence of the semi-discrete form are discussed. Then, a fully discrete scheme of 2D-STFBTEs is derived and the convergence is investigated. Finally, some numerical examples based on linear piecewise polynomials and quadratic piecewise polynomials are given to prove the correctness of our theoretical analysis.