Finite Volume Element Methods for Two-Dimensional Time Fractional Reaction-Diffusion Equations on Triangular Grids

TL;DR

This paper develops finite volume element methods for solving two-dimensional time fractional reaction-diffusion equations on triangular grids, providing stability, error estimates, and numerical validation.

Contribution

It introduces a novel analysis technique to establish error estimates in $H^1$-norm for FVE schemes applied to fractional PDEs.

Findings

Proves existence and uniqueness of the fully discrete scheme

Derives stability and optimal $L^2$-norm error estimates

Numerical examples confirm the method's effectiveness

Abstract

In this paper, the time fractional reaction-diffusion equations with the Caputo fractional derivative are solved by using the classical $L1$-formula and the finite volume element (FVE) methods on triangular grids. The existence and uniqueness for the fully discrete FVE scheme are given. The stability result and optimal \textit{a priori} error estimate in $L^2(\Omega)$-norm are derived, but it is difficult to obtain the corresponding results in $H^1(\Omega)$-norm, so another analysis technique is introduced and used to achieve our goal. Finally, two numerical examples in different spatial dimensions are given to verify the feasibility and effectiveness.