L1 scheme on graded mesh for subdiffusion equation with nonlocal diffusion term

TL;DR

This paper introduces a numerical method combining L1 scheme on graded meshes, finite element, and Newton's methods to solve time fractional diffusion equations with nonlocal terms, addressing initial singularities.

Contribution

It develops a comprehensive numerical approach with error analysis for fractional diffusion equations with nonlocal diffusion, including well-posedness and convergence results.

Findings

Error estimates in L2 and H1 norms validate the method.

Numerical experiments confirm theoretical accuracy and stability.

The approach effectively handles initial singularities in fractional PDEs.

Abstract

The solution of time fractional partial differential equations in general exhibit a weak singularity near the initial time. In this article we propose a method for solving time fractional diffusion equation with nonlocal diffusion term. The proposed method comprises L1 scheme on graded mesh, finite element method and Newton's method. We discuss the well-posedness of the weak formulation at discrete level and derive \emph{a priori} error estimates for fully-discrete formulation in $L^2(\Omega)$ and $H^1(\Omega)$ norms. Finally, some numerical experiments are conducted to validate the theoretical findings.