A note on a priori $$\mathbf {L^p}$$ L p -error estimates for the obstacle problem

Abstract

This paper is concerned with a priori error estimates for the piecewise linear finite element approximation of the classical obstacle problem. We demonstrate by means of two one-dimensional counterexamples that the \(L^2\)-error between the exact solution u and the finite element approximation \(u_h\) is typically not of order two even if the exact solution is in \(H^2(\varOmega )\) and an estimate of the form \(\Vert u - u_h\Vert _{H^1} \le {Ch}\) holds true. This shows that the classical Aubin–Nitsche trick which yields a doubling of the order of convergence when passing over from the \(H^1\)- to the \(L^2\)-norm cannot be generalized to the obstacle problem.