Maximum norm error estimators for three-dimensional elliptic problems

In this paper we define an a posteriori error estimator for finite element approximations of 3-d elliptic problems. We prove that the estimator is equ ivalent, up to logarithmic factors of the meshsize, to the maximum norm of the error. The results are valid for an arbitrary polyhedral domain and rat her general meshes. We also obtain analogous results for the nonconforming method of Crouzeix-Raviart. Finally, we present some numerical results comp aring adaptive procedures based on controlling the error in different norms .