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On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H 1-Stability of L 2-Projection

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An Erratum to this article was published on 28 September 2015

Abstract

Newest vertex bisection (NVB) is a popular local mesh-refinement strategy for regular triangulations that consist of simplices. For the 2D case, we prove that the mesh-closure step of NVB, which preserves regularity of the triangulation, is quasi-optimal and that the corresponding L 2-projection onto lowest-order Courant finite elements (P1-FEM) is always H 1-stable. Throughout, no additional assumptions on the initial triangulation are imposed. Our analysis thus improves results of Binev et al. (Numer. Math. 97(2):219–268, 2004), Carstensen (Constr. Approx. 20(4):549–564, 2004), and Stevenson (Math. Comput. 77(261):227–241, 2008) in the sense that all assumptions of their theorems are removed. Consequently, our results relax the requirements under which adaptive finite element schemes can be mathematically guaranteed to convergence with quasi-optimal rates.

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Acknowledgements

The research of the authors M. Karkulik and D. Praetorius is supported through the FWF project Adaptive Boundary Element Method, funded by the Austrian Science Fund (FWF) under grant P21732, see http://www.asc.tuwien.ac.at/abem.

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Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Santiago, Chile

    Michael Karkulik

  2. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8-10, 1040, Vienna, Austria

    David Pavlicek & Dirk Praetorius

Authors
  1. Michael Karkulik
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  2. David Pavlicek
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  3. Dirk Praetorius
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Correspondence to Michael Karkulik.

Additional information

Communicated by Wolfgang Dahmen.

Dedicated to Carsten Carstensen on the occasion of his 50th birthday.

An erratum to this article is available at http://dx.doi.org/10.1007/s00365-015-9309-z.

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Karkulik, M., Pavlicek, D. & Praetorius, D. On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H 1-Stability of L 2-Projection. Constr Approx 38, 213–234 (2013). https://doi.org/10.1007/s00365-013-9192-4

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  • DOI: https://doi.org/10.1007/s00365-013-9192-4

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