LAUSR

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 $$L^2$$L2-error between the exact solution u and the finite element approximation $$u_h$$uh is typically not of order two even if the exact solution is in $$H^2(\varOmega )$$H2(Ω) and an estimate of the form $$\Vert u - u_h\Vert _{H^1} \le {Ch}$$‖u-uh‖H1≤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 $$H^1$$H1- to the $$L^2$$L2-norm cannot be generalized to the obstacle problem.