Error analysis of two-grid virtual element method for nonlinear parabolic problems on general polygonal meshes

Abstract

In this paper, we present a two-grid virtual element method to solve the nonlinear parabolic problem. The nonlinear terms $f\left(u\right)$ are approximated by using the $L^2$ orthogonal projection, and the fine-grid discrete form is enhanced by Newton iteration. We first prove the $H^1$-norm error estimate for the fully discrete problem. Furthermore, the a priori error estimates of two-grid method in the $L^2$- and $H^1$-norms achieve the optimal order $O(h^{k+1}+H^{2k}+\tau )$ and $O(h^k+H^{2k}+\tau )$, respectively. Finally, we used two numerical examples to validate our two-grid algorithm, which is consistent with our theoretical results.