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Pointwise a posteriori error control for elliptic obstacle problems

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Abstract

We consider a finite element method for the elliptic obstacle problem over polyhedral domains in ℝd, which enforces the unilateral constraint solely at the nodes. We derive novel optimal upper and lower a posteriori error bounds in the maximum norm irrespective of mesh fineness and the regularity of the obstacle, which is just assumed to be Hölder continuous. They exhibit optimal order and localization to the non-contact set. We illustrate these results with simulations in 2d and 3d showing the impact of localization in mesh grading within the contact set along with quasi-optimal meshes.

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Authors and Affiliations

  1. Department of Mathematics and Institute of Physical Science and Technology, University of Maryland, College Park, MD, 20742, USA

    Ricardo H. Nochetto

  2. Institut für Angewandte Mathematik, Hermann-Herder-Str. 10, 79104, Freiburg, Germany

    Kunibert G. Siebert

  3. Dipartimento di Matematica, Università degli Studi di Milano, Via C. Saldini 50, 20133, Milano, Italy

    Andreas Veeser

Authors
  1. Ricardo H. Nochetto
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  2. Kunibert G. Siebert
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  3. Andreas Veeser
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Corresponding author

Correspondence to Ricardo H. Nochetto.

Additional information

Partially supported by NSF Grant DMS-9971450 and NSF/DAAD Grant INT-9910086.

Partially suported by DAAD/NSF grant ``Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF''.

Partially supported by DAAD/NSF grant ``Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF'', and by the TMR network ``Viscosity solutions and their Applications'', Italian M.I.U.R. projects ``Scientific Computing: Innovative Models and Numerical Methods'' and ``Symmetries, Geometric Structures, Evolution and Memory in PDEs''.

Mathematics Subject Classification (1991): 65N15, 65N30, 35J85

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Nochetto, R., Siebert, K. & Veeser, A. Pointwise a posteriori error control for elliptic obstacle problems. Numer. Math. 95, 163–195 (2003). https://doi.org/10.1007/s00211-002-0411-3

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  • DOI: https://doi.org/10.1007/s00211-002-0411-3

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