A two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes

Abstract

In this paper, we develop a two-grid virtual element method for nonlinear variable-order time-fractional diffusion equation on polygonal meshes. The L1 graded mesh scheme is considered in the time direction, and the VEM is used to approximate spatial direction. The two-grid virtual element algorithm reduces the solution of the nonlinear time fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithm not only saves total computational cost, but also maintains the optimal accuracy. Optimal $L^2$ error estimates are analysed in detail for both the VEM scheme and the corresponding two-grid VEM scheme. Finally, numerical experiments presented confirm the theoretical findings.

Keywords:

• Virtual element method
• nonlinear
• variable-order fractional equation
• two-grid
• polygonal meshes
• a priori error estimate

Mathematics subject classifications:

• 65M60
• 65N30
• 34K37
• 65M15
• 65M55

Disclosure statement

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

Additional information

Funding

This work is supported by the State Key Program of National Natural Science Foundation of China [grant number 11931003] and National Natural Science Foundation of China [grant number 41974133], Hunan Provincial Innovation Foundation for Postgraduate, China [grant number XDCX2021B098], Postgraduate Scientific Research Innovation Project of Hunan Province [grant number CX20210597].