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Partial Differential Equations and Functional Analysis pp 27–43Cite as

Clément Interpolation and Its Role in Adaptive Finite Element Error Control

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Part of the Operator Theory: Advances and Applications book series (OT,volume 168)

Abstract

Several approximation operators followed Philippe Clément’s seminal paper in 1975 and are hence known as Clément-type interpolation operators, weak-, or quasi-interpolation operators. Those operators map some Sobolev space V ⊂ W k,p(Ω) onto some finite element space V h ⊂ W k,p(Ω) and generalize nodal interpolation operators whenever W k,p(Ω) ⊄ C 0(Ω), i.e., when p ≤ n/k for a bounded Lipschitz domain Ω ⊂ ℝn. The original motivation was H 2 ⊄ C 0(Ω) for higher dimensions n ≥ 4 and hence nodal interpolation is not well defined.

Todays main use of the approximation operators is for a reliability proof in a posteriori error control. The survey reports on the class of Clément type interpolation operators, its use in a posteriori finite element error control and for coarsening in adaptive mesh design.

Keywords

  • Posteriori Error
  • Error Control
  • Posteriori Error Estimation
  • Interpolation Operator
  • Dual Norm

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In honor of the retirement of Philippe Clàment.

The author is supported by the DFG Research Center Matheon “Mathematics for key technologies” in Berlin.

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Authors and Affiliations

  1. Humboldt-Universität zu Berlin, Unter den Linden 6, D-10099, Berlin, Germany

    Carsten Carstensen

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  1. Carsten Carstensen
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Editors and Affiliations

  1. Technische Universiteit Delft DIAM, P.O. Box 5031, 2600 GA, Delft, The Netherlands

    Erik Koelink, Jan van Neerven, Ben de Pagter & Guido Sweers, ,  & 

  2. Institut für Analysis und Scientific Computing, Technische Universität Wien, Wiedner Hauptstrasse 8-10 / 101, 1040, Wien, Austria

    Annemarie Luger & Harald Woracek & 

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Carstensen, C. (2006). Clément Interpolation and Its Role in Adaptive Finite Element Error Control. In: Koelink, E., van Neerven, J., de Pagter, B., Sweers, G., Luger, A., Woracek, H. (eds) Partial Differential Equations and Functional Analysis. Operator Theory: Advances and Applications, vol 168. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7601-5_2

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