The virtual element method for solving two-dimensional fractional cable equation on general polygonal meshes

Abstract

In this paper, the conforming virtual element method (VEM) is considered to solve the two-dimensional fractional cable equation involving two Riemann–Liouville fractional derivatives. We adopt the Backward Euler Method and the classical $L1$ scheme for the numerical discrete scheme of the time derivative. Meanwhile, the conforming VEM, which is generated for arbitrary order of accuracy and the arbitrary polygonal meshes, is analysed for the discretization of the spatial direction. Based on the energy projection operator, the fully discrete formula is proved to be unconditionally stable, and the optimal convergence results are derived with regard to the $L^2$-norm in detail. Finally, some numerical experiments are implemented to verify the theoretical results.

Keywords:

• Virtual element method
• fractional cable equation
• Riemann–Liouville fractional derivative
• polygonal meshes

Mathematic Subject classifications:

• 03-08
• 26A33
• 35R11

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work is supported by National Natural Science Foundation of China [grant numbers 11931003, 41974133, 12271233]. Hunan Provincial Innovation Foundation for Postgraduate, China [grant number XDCX2023Y119], Postgraduate Scientific Research Innovation Project of Hunan Province [grant number CX20230620].