Finite difference/Galerkin finite element methods for a fractional heat conduction-transfer equation

In this paper, we develop and compare four kinds of fully discrete continuous Galerkin finite element methods for a fractional heat transport equation describing the deep underground unsteady flow. We adopt continuous Galerkin finite element methods for the spatial discretization and the backward Euler together with linear interpolation (L1), second-order backward differentiation formula (BDF2), weighted and shifted Grunwald difference (WSGD), and Crank-Nicolson (CN) time-stepping methods for time discretization. We derive the stability estimates and a priori error estimates for the semidiscrete and fully discrete finite element schemes. Finally, numerical comparisons of the four families of fully discrete finite element methods are presented to illustrate the effectiveness of the studied numerical schemes.