A continuous Petrov–Galerkin method for time-fractional Fokker–Planck equation

A continuous Petrov–Galerkin (CPG) method for the time discretization of a time-fractional Fokker–Planck equation with a general driving force involving fractional exponent $\alpha \in (0,1)$ is proposed. An $\alpha$-robust stability bound for the time discrete solution is obtained for $\alpha \in (1/2,1)$. Regarding error analysis, time-graded meshes are utilized to address the singular behavior of the continuous solution near origin. We present the nodal error estimate with non-uniform time meshes and obtain the optimal second-order accurate estimate for $\alpha \in (1/2,1)$. The fully-discrete scheme by employing standard (continuous) finite element method in space is considered, and the corresponding error is estimated. Numerical experiments demonstrate that the assumptions of time-graded meshes can be further relaxed, and the results exhibit second-order accuracy for $\alpha \in (0,1)$.