Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

Abstract

The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the $L^2$-convergence of the scheme are proved for all variable-order $\alpha (t)\in (0,1)$. The proposed method is of accuracy-order $O(\tau ^3+h^{k+1})$ , where $\tau$, h, and k are the temporal step size, the spatial step size, and the degree of piecewise $P^k$ polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

Funding

Foundation of Henan Educational Committee(19A110005)