Abstract
For the linear finite element solution to a linear elliptic model problem, we derive an error estimator based upon appropriate gradient recovery by local averaging. In contrast to popular variants like the ZZ estimator, our estimator contains some additional terms that ensure reliability also on coarse meshes. Moreover, the enhanced estimator is proved to be (locally) efficient and asymptotically exact whenever the recovered gradient is superconvergent. We formulate an adaptive algorithm that is directed by this estimator and illustrate its aforementioned properties, as well as their importance, in numerical tests.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Ainsworth, M., Craig, A.W.: A posteriori error estimators in the finite element method. Numer. Math. 60, 429–463 (1992)
Ainsworth, M., Oden, J.T.: A posteriori error estimation in finite element analysis. Pure and Applied Mathematics (New York), Wiley-Interscience [John Wiley & Sons], New York, 2000
Babuška, I., Strouboulis, T.: The finite element method and its reliability. Numerical Mathematics and Scientific Computation, The Clarendon Press Oxford University Press, New York, 2001.
Bank, R.E., Xu, J.: Asymptotically exact a posteriori error estimators. II. General unstructured grids. SIAM J. Numer. Anal. 41, 2313–2332 (2003) (electronic)
Bänsch, E., Morin, P., Nochetto, R.H.: An adaptive Uzawa FEM for the Stokes problem: convergence without the inf-sup condition. SIAM J. Numer. Anal. 40, 1207–1229 (2002) (electronic)
Bernardi, C., Verfürth, R.: Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85, 579–608 (2000)
Brenner, S.C., Scott, L.R.: The mathematical theory of finite element methods. vol. 15 of Texts in Applied Mathematics, Springer, New York, 1994
Carstensen, C.: All first-order averaging techniques for a posteriori finite element error control on unstructured grids are efficient and reliable. Math. Comp. 73, 1153–1165 (2004) (electronic)
Carstensen, C., Bartels, S.: Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. I. Low order conforming, nonconforming, and mixed FEM. Math. Comp. 71, 945–969 (2002) (electronic)
Carstensen, C., Funken, S.A.: Fully reliable localized error control in the FEM. SIAM J. Sci. Comput., 21 1465–1484, (1999/00) (electronic)
Chen, Z., Wu, H.: Uniform convergence of multigrid V-cycle on adaptively refined finite element meshes for second order elliptic problems. Preprint No. 2003–07, Institute of Computational Mathematics, Chinese Academy of Sciences, Beijing, China, http://icmsec.cc.ac.cn/2003research_report.html
Deuflhard, P., Weiser, M.: Global inexact Newton multilevel FEM for nonlinear elliptic problems. In: Multigrid methods V (Stuttgart, 1996), vol. 3 of Lect. Notes Comput. Sci. Eng., Springer, Berlin, 1998, pp. 71–89
Dörfler, W.: A convergent adaptive algorithm for Poisson's equation. SIAM J. Numer. Anal. 33, 1106–1124 (1996)
Faermann, B.: Localization of the Aronszajn-Slobodeckij norm and application to adaptive boundary element methods. II. The three-dimensional case. Numer. Math. 92, 467–499 (2002)
Fierro, F., Veeser, A.: A posteriori error estimators for regularized total variation of characteristic functions. SIAM J. Numer. Anal. 41, 2032–2055 (2003) (electronic)
Hackbusch, W.: Elliptic differential equations. Theory and numerical treatment. vol. 18 of Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 1992
Luce, R., Wohlmuth, B.I.: A local a posteriori error estimator based on equlibrated fluxes. SIAM J. Numer. Anal. 42, 1394–1414 (2004)
Morin, P., Nochetto, R.H., Siebert, K.G.: Data oscillation and convergence of adaptive FEM. SIAM J. Numer. Anal. 38, 466–488 (2000) (electronic)
Morin, P., Nochetto, R.H., Siebert, K.G.: Local problems on stars: a posteriori error estimators, convergence, and performance. Math. Comp. 72, 1067–1097 (2003) (electronic)
Rodríguez, R.: A posteriori error analysis in the finite element method. In: Finite element methods (Jyväskylä, 1993), vol. 164 of Lecture Notes in Pure and Appl. Math., Dekker, New York, 1994, pp. 389–397
Sacchi, R., Veeser, A.: Locally efficient and reliable a posteriori error estimators for Dirichlet problems. Math. Mod. Meth. Appl. Sci. 16, 319–346 (2006)
Schmidt, A., Siebert, K.G.: Design of adaptive finite element software: the finite element toolbox ALBERTA. no. 42 in Lecture Notes in Computational Sciences and Engineering, Springer, 2004
Scott, L.R., Zhang, S.: Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54, 483–493 (1990)
Strouboulis, T., Babuška, I., Gangaraj, S.K.: Guaranteed computable bounds for the exact error in the finite element solution. II. Bounds for the energy norm of the error in two dimensions. Internat. J. Numer. Methods Engrg. 47, 427–475 (2000) Richard H. Gallagher Memorial Issue
Veeser, A.: Convergent adaptive finite elements for the nonlinear Laplacian. Numer. Math. 92, 743–770 (2002)
Verfürth, R.: A review of a posteriori error estimation and adaptive mesh-refinement techniques. Adv. Numer. Math., John Wiley, Chichester, UK, 1996
Xu, J.: An introduction to multilevel methods. In: Wavelets, multilevel methods and elliptic PDEs (Leicester, 1996), Numer. Math. Sci. Comput., Oxford Univ. Press, New York, 1997, pp. 213–302
Xu, J., Zhang, Z.: Analysis of recovery type a posteriori error estimators for mildly structured grids. Math. Comp. 73, 1139–1152 (2004) (electronic)
Zienkiewicz, O.C., Zhu, J.Z.: A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Eng. 24, 337–357 (1987)
Author information
Authors and Affiliations
Corresponding author
Additional information
Research partially supported by Italian MIUR Cofin 2003 ``Modellistica numerica per il calcolo scientifico e applicazioni avanzate'' and Cofin 2004 ``Metodi numerici avanzati per equazioni alle derivate parziali di interesse applicativo''.
Rights and permissions
About this article
Cite this article
Fierro, F., Veeser, A. A posteriori error estimators, gradient recovery by averaging, and superconvergence. Numer. Math. 103, 267–298 (2006). https://doi.org/10.1007/s00211-005-0671-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00211-005-0671-9