Galerkin finite element method for higher dimensional multi-term fractional diffusion equation on non-uniform meshes

Abstract

In this paper, we study Galerkin finite element methods for a class of higher dimensional multi-term fractional diffusion equations. The finite difference approximation of Caputo derivative on non-uniform meshes is used in temporal direction and Galerkin finite element method is used in spatial direction. We prove that semi-discrete and fully discrete finite element schemes are unconditionally stable. Meanwhile, -norm convergence properties of the two schemes are proved rigorously. To confirm our theoretical analysis, we give some numerical examples in both two-dimensional (2D) and three-dimensional (3D) spaces. Finally, a moving local refinement technique in temporal direction is used to improve the accuracy of numerical solution.

Keywords:

• Fractional diffusion equation
• non-uniform meshes
• Galerkin finite element method
• stability analysis
• error estimates

AMS Subject Classifications:

• 97N40
• 65D07
• 74S05
• 26A33
• 34A08
• 65L60

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by NSF of China [grant number 11371157]; Natural Science Foundation of Anhui Higher Education Institutions of China [grant number KJ2016A492]; Natural Science Foundation of Bozhou College [grant number BSKY201426].