Remember Me
Or use your Academic/Social account:


Or use your Academic/Social account:


You have just completed your registration at OpenAire.

Before you can login to the site, you will need to activate your account. An e-mail will be sent to you with the proper instructions.


Please note that this site is currently undergoing Beta testing.
Any new content you create is not guaranteed to be present to the final version of the site upon release.

Thank you for your patience,
OpenAire Dev Team.

Close This Message


Verify Password:
Verify E-mail:
*All Fields Are Required.
Please Verify You Are Human:
fbtwitterlinkedinvimeoflicker grey 14rssslideshare1
Ferraz-Leite, S.; Ortner, Christoph; Praetorius, Dirk (2008)
Publisher: Springer
Languages: English
Types: Article
Subjects: QA
We discuss several adaptive mesh-refinement strategies based on (h − h/2)-error estimation. This class of adaptivemethods 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.
  • The results below are discovered through our pilot algorithms. Let us know how we are doing!

    • 1. Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Wiley-Interscience [John Wiley & Sons], New-York (2000)
    • 2. Bänsch, E.: Local mesh refinement in 2 and 3 dimensions. IMPACT Comput. Sci. Eng. 3, 181- 191 (1991)
    • 3. Bank, R.: Hierarchical bases and the finite element method. Acta Numerica 5, 1-45 (1996)
    • 4. Bank, R., Smith, R.: A posteriori error-estimates based on hierarchical bases. SIAM J. Numer. Anal. 30, 921-935 (1993)
    • 5. Bank, R., Weiser, A.: Some a posteriori error estimators for elliptic partial differential equations. Math. Comp. 44, 283-301 (1985)
    • 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)
    • 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)
    • 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)
    • 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)
    • 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)
    • 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)
    • 12. Dörfler, W.: A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33, 1106- 1124 (1996)
    • 13. Dörfler, W., Nochetto, R.: Small data oscillation implies the saturation assumption. Numer. Math. 91, 1-12 (2002)
    • 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)
    • 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)
    • 16. Hairer, E., Nørsett, S., Wanner, G.: Solving ordinary differential equations I. Nonstiff problems. Springer, New York (1987)
    • 17. Kossaczky, I.: A recursive approach to local mesh refinement in two and three dimensions. J. Comput. Appl. Math. 55, 275-288 (1995)
    • 18. McLean, W.: Strongly elliptic systems and boundary integral equations. Cambridge University Press, Cambridge (2000)
    • 19. Morin, P., Nochetto, R., Siebert, K.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466-488 (2000)
    • 20. Sauter, S., Schwab, C.: Randelementmethoden: Analyse, Numerik und Implementierung schneller Algorithmen. Teubner Verlag, Wiesbaden (2004)
    • 21. Sewell, E.: Automatic generation of triangulations for piecewise polynomial approximations. Ph.D. thesis, Purdue University, West Lafayette (1972)
    • 22. Verfürth, R.: A posteriori error estimation and adaptive mesh refinement techniques. J. Comput. Appl. Math. 50, 67-83 (1994)
    • 23. Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Teubner, Stuttgart (1996)
  • No related research data.
  • No similar publications.

Share - Bookmark

Funded by projects

Cite this article