A fast second order PDE approach for the space-time fractional parabolic problems

We study a fast second-order PDE approach for solving the space-time parabolic equations with fractional diffusion and Caputo fractional time derivative. To localize the space fractional elliptic operator, we map a Dirichlet boundary condition to a Neumann condition via an extension problem on the semi-infinite cylinder. For the equivalent extension problem, we use the fast L2-1 ${}_{\sigma }$ method based on the sum-of-exponentials to speed up the evaluation of the time fractional Caputo derivative, and the tensor product finite element method to discretize the spatial direction on the truncated cylinder domain. Then, the stability and $\alpha$-robust error estimates of the fully discrete scheme are derived. Finally, the numerical experiments are presented to demonstrate the effectiveness of our scheme.