Numerische Mathematik

, Volume 116, Issue 2, pp 291–316 | Cite as

Convergence of simple adaptive Galerkin schemes based on hh/2 error estimators

  • S. Ferraz-Leite
  • C. Ortner
  • D. PraetoriusEmail author
Open Access


We discuss several adaptive mesh-refinement strategies based on (hh/2)-error estimation. This class of adaptive methods is particularly popular in practise since it is problem independent and requires virtually no implementational overhead. We prove that, under the saturation assumption, these adaptive algorithms are convergent. Our framework applies not only to finite element methods, but also yields a first convergence proof for adaptive boundary element schemes. For a finite element model problem, we extend the proposed adaptive scheme and prove convergence even if the saturation assumption fails to hold in general.

Mathematics Subject Classification (2000)

65N30 65N38 65N50 65N12 



Parts of the results have been achieved during a research stay of C.O. and D.P. at the Hausdorff Institute for Mathematics in Bonn, which is thankfully acknowledged. S.F. acknowledges a grant of the graduate school Differential Equations – Models in Science and Engineering, funded by the Austrian Science Fund (FWF) under grant W800-N05. D.P. is partially supported through the research project Adaptive Boundary Element Method, funded by the Austrian Science Fund (FWF) under grant P21732.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Ainsworth M., Oden J.T.: A posteriori error estimation in finite element analysis. Wiley-Interscience [John Wiley & Sons], New-York (2000)zbMATHGoogle Scholar
  2. 2.
    Bänsch E.: Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Eng. 3, 181–191 (1991)zbMATHCrossRefGoogle Scholar
  3. 3.
    Bank R.: Hierarchical bases and the finite element method. Acta Numerica 5, 1–45 (1996)CrossRefMathSciNetGoogle Scholar
  4. 4.
    Bank R., Smith R.: A posteriori error-estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921–935 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bank R., Weiser A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283–301 (1985)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Bornemann F., Erdmann B., Kornhuber R.: A-posteriori error-estimates for elliptic problems in 2 and 3 space dimensions. SIAM J. Numer. Anal. 33, 1188–1204 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Carstensen C., Faermann B.: Mathematical foundation of a posteriori error estimates and adaptive mesh-refining algorithms for boundary integral equations of the first kind. Eng. Anal. Bound. Elem. 25, 497–509 (2001)zbMATHCrossRefGoogle Scholar
  8. 8.
    Carstensen C., Maischak M., Praetorius D., Stephan E.P.: Residual-based a posteriori error estimate for hypersingular equation on surfaces. Numer. Math. 97(3), 397–425 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Carstensen C., Praetorius D.: Averaging techniques for the effective numerical solution of Symm’s integral equation of the first kind. SIAM J. Sci. Comput. 27, 1226–1260 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Cascon J., Kreuzer C., Nochetto R., Siebert K.: Quasi-optimal convergence rate for an adaptive finite element method. SIAM J. Numer. Anal. 46, 2524–2550 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Deuflhard P., Leinen P., Yserentant H.: Concepts of an adaptive hierarchical finite element code. IMPACT Comput. in. Sci. and Eng. 1, 3–35 (1989)zbMATHCrossRefGoogle Scholar
  12. 12.
    Dörfler W.: A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Dörfler W., Nochetto R.: Small data oscillation implies the saturation assumption. Numer. Math. 91, 1–12 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Ferraz-Leite S., Praetorius D.: Simple a posteriori error estimators for the h-version of the boundary element method. Computing 83, 135–162 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Graham I., Hackbusch W., Sauter S.: Finite elements on degenerate meshes: inverse-type inequalities and applications. IMA J. Numer. Anal. 25, 379–407 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Hairer E., Nørsett S., Wanner G.: Solving ordinary differential equations I. Nonstiff problems. Springer, New York (1987)zbMATHGoogle Scholar
  17. 17.
    Kossaczky I.: A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55, 275–288 (1995)CrossRefMathSciNetGoogle Scholar
  18. 18.
    McLean W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  19. 19.
    Morin P., Nochetto R., Siebert K.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Sauter S., Schwab C.: Randelementmethoden: Analyse, Numerik und Implementierung schneller Algorithmen. Teubner Verlag, Wiesbaden (2004)zbMATHGoogle Scholar
  21. 21.
    Sewell, E.: Automatic generation of triangulations for piecewise polynomial approximations. Ph.D. thesis, Purdue University, West Lafayette (1972)Google Scholar
  22. 22.
    Verfürth R.: A posteriori error estimation and adaptive mesh refinement techniques. J. Comput. Appl. Math. 50, 67–83 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Verfürth R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner, Stuttgart (1996)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2010

Open AccessThis is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (, which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

Authors and Affiliations

  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria
  2. 2.Mathematical InstituteUniversity of OxfordOxfordUK

Personalised recommendations