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 (t) ∈ (0, 1). The proposed method is of accuracyorder O(3 + h k+1) , where , 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.

An L1 type difference/Galerkin spectral scheme for variable-order time-fractional nonlinear diffusion–reaction equations with fixed delay

“…Recently, there are several investigations on numerical schemes to approximate the solutions of variable-order time-fractional diffusion equations in the literature. We mention here the works by Tavares et al [29], Chen et al [30], Karniadakis et al [31][32][33], Zheng et al [34][35][36], Pang and Sun [37], Wei et al [38,39] and Liu et al [40]. Nevertheless, rigorous numerical and mathematical analysis of variable-order fractional differential equations remains wide-open.…”