Optimal maximum norm estimates for virtual element methods

TL;DR

This paper derives optimal maximum norm error estimates for high-order virtual element methods, demonstrating their convergence properties and validating results through numerical experiments on polygonal meshes.

Contribution

It establishes the first optimal maximum norm error estimates for virtual element methods using advanced regularity and Green's function analysis.

Findings

Optimal convergence in maximum norm for virtual element methods.

Validation of theoretical results through numerical experiments.

High-order virtual elements achieve optimal $L^{}$ norm convergence.

Abstract

The maximum norm error estimations for virtual element methods are studied. To establish the error estimations, we prove higher local regularity based on delicate analysis of Green's functions and high-order local error estimations for the partition of the virtual element solutions. The maximum norm of the exact gradient and the gradient of the projection of the virtual element solutions are proved to achieve optimal convergence results. For high-order virtual element methods, we establish the optimal convergence results in $L^{\infty}$ norm. Our theoretical discoveries are validated by a numerical example on general polygonal meshes.