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.

翻译：对虚拟元素方法的最大标准误差估计进行了研究。为了确定误差估计,根据对Green函数的微妙分析和对虚拟元素解决方案分割的高分级本地误差估计,我们证明地方的规律性更高。精确梯度和虚拟元素解决方案预测的梯度的最大标准证明能够取得最佳趋同结果。对于高阶虚拟元素方法,我们用$L ⁇ infty}标准来确定最佳趋同结果。我们的理论发现通过一般多边模类的数字示例来验证。