$$\mathbf {L^\infty }$$ L ∞ -error estimates for the obstacle problem revisited

Abstract

In this paper, we present an alternative approach to a priori \(L^\infty \)-error estimates for the piecewise linear finite element approximation of the classical obstacle problem. Our approach is based on stability results for discretized obstacle problems and on error estimates for the finite element approximation of functions under pointwise inequality constraints. As an outcome, we obtain the same order of convergence proven in several works before. In contrast to prior results, our estimates can, for example, also be used to study the situation where the function space is discretized but the obstacle is not modified at all.