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

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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.

Mathematics Subject Classification

65K15 65N15 65N30 

Notes

Acknowledgements

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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