Fast and efficient finite difference/finite element method for the two-dimensional multi-term time-space fractional Bloch-Torrey equation

Abstract This paper proposes a numerical approximation combining finite difference method in time and finite element method in space to solve the two-dimensional time-space fractional Bloch-Torrey equation. Unlike the existing works which focus on the decoupled model, the proposed numerical scheme is established and discussed based on the coupled equations which is from the separation of real and imaginary parts of the original fractional Bloch-Torrey model. In addition, in order to get fast estimation to Caputo fractional derivative and reduce the storage of numerical scheme, an efficient sum-of-exponentials approximation for the kernel t − α , α ∈ ( 0 , 1 ) is adopted. Under the non-uniform time mesh, the stability and convergence are discussed for the semi-discrete scheme, and the error estimate is investigated for the fully discrete scheme in detail. Finally, several numerical tests are provided to verify the correctness of the obtained theoretical results and the effectiveness of our method.