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

Zhen Wang

doi:10.1007/s42967-023-00257-x

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

Commun Appl Math Comput. 2023 Apr 11:1-25. doi: 10.1007/s42967-023-00257-x. Online ahead of print.

Author

Zhen Wang¹

Affiliation

¹ School of Mathematical Sciences, Jiangsu University, Zhenjiang, 212013 Jiangsu China.

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 $α$ -robust using the newly established Gronwall inequalities, that is, it remains valid when $α \to 1^{-}$ . Numerical experiments are given to demonstrate the theoretical statements.

Keywords: Caputo-Hadamard derivative; Discrete Gronwall inequality; Error estimate; L1 formula; Local discontinuous Galerkin (LDG) method.

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