Numerische Mathematik

, Volume 139, Issue 1, pp 27–45 | Cite as

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



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.

Mathematics Subject Classification

65K15 65N15 65N30 



We would like to thank the two anonymous reviewers for their helpful suggestions and comments.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Faculty of MathematicsTU DortmundDortmundGermany

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