SIAM Journal on Numerical Analysis, Volume 60, Issue 3, Page 1251-1280, June 2022.
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.

中文翻译：

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虚拟元素方法的最优最大范数估计

SIAM 数值分析杂志，第 60 卷，第 3 期，第 1251-1280 页，2022 年 6 月
。研究了虚拟元素方法的最大范数误差估计。为了建立误差估计，我们基于对格林函数的精细分析和虚拟元素解的划分的高阶局部误差估计证明了更高的局部规律性。证明了精确梯度的最大范数和虚拟元解的投影梯度可以达到最优的收敛结果。对于高阶虚元方法，我们在 $L^{\infty}$ 范数中建立了最优收敛结果。我们的理论发现通过一个关于一般多边形网格的数值例子得到验证。