Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Local problems on stars: A posteriori error estimators, convergence, and performance


Authors: Pedro Morin, Ricardo H. Nochetto and Kunibert G. Siebert
Journal: Math. Comp. 72 (2003), 1067-1097
MSC (2000): Primary 65N12, 65N15, 65N30, 65N50, 65Y20
DOI: https://doi.org/10.1090/S0025-5718-02-01463-1
Published electronically: November 7, 2002
MathSciNet review: 1972728
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A new computable a posteriori error estimator is introduced, which relies on the solution of small discrete problems on stars. It exhibits built-in flux equilibration and is equivalent to the energy error up to data oscillation without any saturation assumption. A simple adaptive strategy is designed, which simultaneously reduces error and data oscillation, and is shown to converge without mesh pre-adaptation nor explicit knowledge of constants. Numerical experiments reveal a competitive performance, show extremely good effectivity indices, and yield quasi-optimal meshes.


References [Enhancements On Off] (What's this?)

  • 1. M. Ainsworth and I. Babuska, Reliable and robust a posteriori error estimation for singularly perturbed reaction-diffusion problems, SIAM J. Numer. Anal., 36 (1999), 331-353. MR 99k:65083
  • 2. M. Ainsworth and J.T. Oden, A unified approach to a posteriori error estimation based on element residual methods, Numer. Math. 65 (1993), 23-50. MR 95a:65185
  • 3. I. Babuska and A. Miller, A feedback finite element method with a posteriori error estimations: Part I. The finite element method and some basic properties of the a posteriori error estimator, Comp. Meth. Appl. Mech. Eng., 61 (1987), 1-40. MR 88d:73036
  • 4. R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), No. 170, 283-301. MR 86g:65207
  • 5. C. Carstensen and S.A. Funken, Fully reliable localised error control in the FEM, SIAM J. Sci. Comp., 21 (1999/00), 1465-1484. MR 2000k:65205
  • 6. W. Dörfler, A convergent adaptive algorithm for Poisson's equation, SIAM J. Numer. Anal., 33 (1996), 1106-1124. MR 97e:65139
  • 7. W. Dörfler and R.H. Nochetto, Small data oscillation implies the saturation assumption, Numer. Math., 91 (2002), 1-12.
  • 8. R.G. Durán and M.A. Muschietti, An explicit right inverse of the divergence operator in $W_0^{1,p}(\Omega)^n$, Studia Math., 148 (2001), 207-219.
  • 9. R.B. Kellogg, On the Poisson equation with intersecting interfaces, Applicable Analysis, 4 (1975), 101-129. MR 52:14623
  • 10. A. Kufner, Weighted Sobolev Spaces, Teubner Texte zur Mathematik, Bd. 31, Teubner, Leibzip 1980. MR 84e:46029
  • 11. P. Morin, R.H. Nochetto and K.G. Siebert, Data Oscillation and Convergence of Adaptive FEM, SIAM J. Numer. Anal., 38 (2000), 466-488. MR 2001g:65157
  • 12. P. Morin, R.H. Nochetto and K.G. Siebert, Basic principles for convergence of adaptive higher-order FEM -- Application to linear elasticity, in preparation.
  • 13. J. Necas, Sur une methode pour resoudre les equations aux derivees partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm. Sup. Pisa 16 (1962), 305-326. MR 29:357
  • 14. R.H. Nochetto Removing the saturation assumption in a posteriori error analysis, Istit. Lombardo Sci. Lett. Rend. A, 127 (1993), 67-82. MR 95c:65187
  • 15. A. Schmidt and K.G. Siebert, ALBERT -- Software for scientific computations and applications, Acta Math. Univ. Comenianae 70 (2001), 105-122.
  • 16. A. Schmidt and K.G. Siebert, ALBERT: An adaptive hierarchical finite element toolbox, Documentation, Preprint 06/2000 Universität Freiburg, 244 p.
  • 17. T. Strouboulis, I. Babuska and S.K. Gangaraj, Guaranteed computable bounds for the exact error in the finite elment solution - Part II: bounds for the energy norm of the error in two dimensions, Int. J. Numer. Meth. Engng. 47 (2000), 427-475. MR 2001a:65109
  • 18. R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner, 1996.

Similar Articles

Retrieve articles in Mathematics of Computation with MSC (2000): 65N12, 65N15, 65N30, 65N50, 65Y20

Retrieve articles in all journals with MSC (2000): 65N12, 65N15, 65N30, 65N50, 65Y20


Additional Information

Pedro Morin
Affiliation: Departamento de Matemática, Facultad de Ingeniería Química, Universidad Nacional del Litoral, Santiago del Estero 2829, 3000 Santa Fe, Argentina
Email: pmorin@math.unl.edu.ar

Ricardo H. Nochetto
Affiliation: Department of Mathematics and Institute for Physical Science and Technology, University of Maryland, College Park, Maryland 20742
Email: rhn@math.umd.edu

Kunibert G. Siebert
Affiliation: Institut für Angewandte Mathematik, Hermann-Herder-Str. 10, 79104 Freiburg, Germany
Email: kunibert@mathematik.uni-freiburg.de

DOI: https://doi.org/10.1090/S0025-5718-02-01463-1
Keywords: A posteriori error estimators, local problems, stars, data oscillation, adaptivity, convergence, performance
Received by editor(s): October 12, 2000
Received by editor(s) in revised form: September 26, 2001
Published electronically: November 7, 2002
Additional Notes: The first author was partially supported by CONICET of Argentina, NSF Grant DMS-9971450, and NSF/DAAD Grant INT-9910086. This work was developed while this author was visiting the University of Maryland
The second author was partially supported by NSF Grant DMS-9971450 and NSF/DAAD Grant INT-9910086
The third author was partially suported by DAAD/NSF grant “Projektbezogene Förderung des Wissenschaftleraustauschs in den Natur-, Ingenieur- und den Sozialwissenschaften mit der NSF”. Part of this work was developed while this author was visiting the University of Maryland
Article copyright: © Copyright 2002 American Mathematical Society

American Mathematical Society