We present a new logarithm free resolvent estimate in maximum-norm for the discrete Laplacian associated with a quasiuniform family of piecewise linear finite element spaces on a convex d-dimensional domain with smooth boundary. We discuss the relation of such an estimate, via semigroup theory, to stablity estimates for associated spatially semidiscrete and fully discrete finite element methods for parabolic equations. The result is joint work with N. Yu. Bakaev and L. B. Wahlbin.