  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.
