Maximum norm resolvent estimates for elliptic finite element operators

We study a finite element approximation A(h), based on simplicial Lagrange elements, of a second order elliptic operator A under homogeneous Dirichlet boundary conditions in two and three dimensions, where h is thought of as a meshsize. The main result of the paper is a new resolvent estimate for the operator A(h) in the L-infinity-norm. This estimate is uniform with respect to h for the case with at least quadratic elements. In the case with linear elements, the estimate contains on the right a factor proportional to (log log 1/n)(nu), where nu = 1 or nu = 5/4 in two or three dimensions, respectively.