Finite volume element methods for two-dimensional time fractional reaction–diffusion equations on triangular grids

Abstract

In this paper, 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 results and optimal a priori error estimate in $L^2(\mathrm{\Omega })$-norm are derived, but it is difficult to obtain the corresponding results in $H^1(\mathrm{\Omega })$-norm, so another analysis technique is introduced and used to achieve our goal. Finally, some numerical results are given to verify the feasibility and effectiveness.

Keywords:

• Time fractional reaction–diffusion equation
• $L1$-formula
• finite volume element method
• stability analysis
• a priori error estimate

2020 Mathematics Subject Classifications:

• 65M08
• 65M15
• 65M60

Acknowledgments

The authors thank the reviewers and editors for their valuable comments and suggestions to improve this work.

Disclosure statement

The authors declare that there is no potential conflict of interest.

Additional information

Funding

This work is supported by the National Natural Science Foundation of China [11701299, 12161063, 11761053], the Natural Science Foundation of Inner Mongolia Autonomous Region [2020MS01003, 2021MS01018], and the Program for Innovative Research Team in Universities of Inner Mongolia Autonomous Region [NMGIRT2207].