L1/LDG Method for Caputo-Hadamard Time Fractional Diffusion Equation.

Abstract

In this paper, a class of discrete Gronwall inequalities is proposed. It is efficiently applied to analyzing the constructed L1/local discontinuous Galerkin (LDG) finite element methods which are used for numerically solving the Caputo-Hadamard time fractional diffusion equation. The derived numerical methods are shown to be $\alpha$-robust using the newly established Gronwall inequalities, that is, it remains valid when $\alpha \to 1^{-}$. Numerical experiments are given to demonstrate the theoretical statements.