Constructive Approximation

, Volume 38, Issue 2, pp 213–234 | Cite as

On 2D Newest Vertex Bisection: Optimality of Mesh-Closure and H1-Stability of L2-Projection

  • Michael Karkulik
  • David Pavlicek
  • Dirk Praetorius
Article

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 L2-projection onto lowest-order Courant finite elements (P1-FEM) is always H1-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.

Keywords

Adaptive finite element methods Regular triangulations Newest vertex bisection L2-Projection H1-Stability 

Mathematics Subject Classification

65N30 65N50 65Y20 

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Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Michael Karkulik
    • 1
  • David Pavlicek
    • 2
  • Dirk Praetorius
    • 2
  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria

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