A Fully Discrete Method for Space-Time Fractional Parabolic Problems

Abstract

This paper considers a parabolic equation involving the Caputo fractional time derivative of order $\gamma\in(0,1)$ and the spectral fractional power, of order $s\in(0,1)$, of a symmetric second order elliptic operator $\mathcal{L}$. By using the Caffarelli–Silvestre extension, the original problem is converted to a quasi-stationary elliptic problem on a semi-infinite cylinder in one more spatial dimension and with a dynamic boundary condition. Based on the solution representation with the Mittag–Leffler function and the eigenpairs of the elliptic operator $\mathcal{L}$, some new regularity results are derived. For the temporal semi-discrete scheme using the Alikhanov difference approximation, stability and convergence are established. Furthermore, the first-degree tensor product finite element method is adopted for the spatial discretization to obtain a fully discrete scheme, and an error estimate is derived. Finally, an efficient algorithm based on a generalized eigenvalue problem is applied to solve the matrix system of the full discretization. Numerical examples are provided to verify the theoretical results.