A conforming virtual element framework for the time-fractional semi-linear reaction–diffusion equation on polygonal meshes

Abstract

This article presents the virtual element method for solving a two-dimensional time-fractional semi-linear reaction–diffusion equation with a fractional derivative of order $\alpha \in (0,1)$ in time. The methodology is based on three fundamental technical components: a fractional version of the Grünwald–Letnikov approximation, a discrete maximal regularity property, and the regularity theory associated with non-linearity. We prove the well-posedness of the discretized scheme developed for the solution of the time-fractional reaction–diffusion equation with a Lipschitz-continuous nonlinear term. The fully discrete scheme inherently maintains stability and consistency by leveraging the discrete maximal regularity and the elliptic projection operator. The convergence in the $L^2$ norm and $H^1$ semi-norm is validated by numerical results over regular Voronoi, distorted hexagons, and non-convex polygon mesh configurations, underlining the practical effectiveness of the proposed scheme. The numerical examples illustrated are important real time applications of the time-fractional nonlinear partial differential equations.