The Use of Finite Difference/Element Approaches for Solving the Time-Fractional Subdiffusion Equation

Abstract

In this paper, two finite difference/element approaches for the time-fractional subdiffusion equation with Dirichlet boundary conditions are developed, in which the time direction is approximated by the fractional linear multistep method and the space direction is approximated by the finite element method. The two methods are unconditionally stable and convergent of order $O(\tau^q+h^{r+1})$ in the $L^2$ norm, where $q=2-\beta$ or 2 when the analytical solution to the subdiffusion equation is sufficiently smooth, $\beta\,(0<\beta<1)$ is the order of the fractional derivative, $\tau$ and $h$ are the step sizes in time and space, respectively, and $r$ is the degree of the polynomial space. The corresponding schemes for the subdiffusion equation with Neumann boundary conditions are presented as well, where the stability and convergence are shown. Numerical examples are provided to verify the theoretical analysis. Comparisons between the algorithms derived in this paper and the existing algorithms are given, which show that our numerical schemes exhibit better performances than the existing ones.