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Averaging techniques yield reliable a posteriori finite element error control for obstacle problems

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Summary.

The reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second-order partial differential equations. This paper establishes related reliable and efficient a posteriori error estimates for the energy-norm error of an obstacle problem on unstructured grids as a model example for variational inequalities. The surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms, consistency terms, and a multiplicative constant.

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Authors and Affiliations

  1. Department of Mathematics, University of Maryland, College Park, MD, 20742, USA

    S. Bartels

  2. Department of Mathematics, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    C. Carstensen

Authors
  1. S. Bartels
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  2. C. Carstensen
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Correspondence to C. Carstensen.

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Mathematics Subject Classification (2000): 35J20, 49J40, 65N30

Acknowledgement The first author (S.B.) thankfully acknowledges partial support by the German Research Foundation (DFG) within the Graduiertenkolleg ‘Effiziente Algorithmen und Mehrskalenmethoden’. The authors thank one anonymous referee for the suggestion of an projected SOR algorithm.

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Bartels, S., Carstensen, C. Averaging techniques yield reliable a posteriori finite element error control for obstacle problems. Numer. Math. 99, 225–249 (2004). https://doi.org/10.1007/s00211-004-0553-6

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  • DOI: https://doi.org/10.1007/s00211-004-0553-6

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