no. 6
Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes
Kunert, Gerd ; Nicaise, Serge1
1 Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise
ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 6, pp. 1013-1043.

We consider a posteriori error estimators that can be applied to anisotropic tetrahedral finite element meshes, i.e. meshes where the aspect ratio of the elements can be arbitrarily large. Two kinds of Zienkiewicz-Zhu (ZZ) type error estimators are derived which originate from different backgrounds. In the course of the analysis, the first estimator turns out to be a special case of the second one, and both estimators can be expressed using some recovered gradient. The advantage of keeping two different analyses of the estimators is that they allow different and partially novel investigations and results. Both rigorous analytical approaches yield the equivalence of each ZZ error estimator to a known residual error estimator. Thus reliability and efficiency of the ZZ error estimation is obtained. The anisotropic discretizations require analytical tools beyond the standard isotropic methods. Particular attention is paid to the requirements on the anisotropic mesh. The analysis is complemented and confirmed by extensive numerical examples. They show that good results can be obtained for a large class of problems, demonstrated exemplary for the Poisson problem and a singularly perturbed reaction diffusion problem.

DOI : 10.1051/m2an:2003065
Classification : 65N15, 65N30, 65N50
Mots clés : anisotropic mesh, error estimator, Zienkiewicz-Zhu estimator, recovered gradient
Kunert, Gerd  ; Nicaise, Serge 1

1 Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise 
@article{M2AN_2003__37_6_1013_0,
     author = {Kunert, Gerd and Nicaise, Serge},
     title = {Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {1013--1043},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {6},
     year = {2003},
     doi = {10.1051/m2an:2003065},
     zbl = {1077.65114},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an:2003065/}
}
TY  - JOUR
AU  - Kunert, Gerd
AU  - Nicaise, Serge
TI  - Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2003
SP  - 1013
EP  - 1043
VL  - 37
IS  - 6
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an:2003065/
DO  - 10.1051/m2an:2003065
LA  - en
ID  - M2AN_2003__37_6_1013_0
ER  - 
%0 Journal Article
%A Kunert, Gerd
%A Nicaise, Serge
%T Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2003
%P 1013-1043
%V 37
%N 6
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an:2003065/
%R 10.1051/m2an:2003065
%G en
%F M2AN_2003__37_6_1013_0
Kunert, Gerd; Nicaise, Serge. Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes. ESAIM: Modélisation mathématique et analyse numérique, Tome 37 (2003) no. 6, pp. 1013-1043. doi : 10.1051/m2an:2003065. http://www.numdam.org/articles/10.1051/m2an:2003065/

[1] M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Wiley (2000). | MR | Zbl

[2] Th. Apel, Anisotropic finite elements: Local estimates and applications, Advances in Numerical Mathematics. Teubner, Stuttgart (1999). | MR | Zbl

[3] I. Babuška, T. Strouboulis and C.S. Upadhyay, A model study of the quality of a posteriori error estimators for linear elliptic problems. Error estimation in the interior of patchwise uniform grids of triangles. Comput. Methods Appl. Mech. Engrg. 114 (1994) 307-378.

[4] I. Babuška, T. Strouboulis, C.S. Upadhyay, S.K. Gangaraj and K. Copps, Validation of a posteriori error estimators by numerical approach. Int. J. Numer. Methods Eng. 37 (1994) 1073-1123. | Zbl

[5] S. Bartels and C. Carstensen, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part II: High order FEM. Math. Comp. 71 (2002) 971-994. | Zbl

[6] J.H. Bramble, J.E. Pasciak and O. Steinbach, On the stability of the L 2 -projection in H 1 (ω). Math. Comp. 71 (2002) 147-156. | Zbl

[7] C. Carstensen, Merging the Bramble-Pasciak-Steinbach and the Crouzeix-Thomée criterion for H 1 -stability of the L 2 -projection onto finite element spaces. Math. Comp. 71 (2002) 157-163. | Zbl

[8] C. Carstensen and S. Bartels, Each averaging technique yields reliable a posteriori error control in FEM on unstructured grids. Part I: Low order conforming, nonconforming, and mixed FEM. Math. Comp. 71 (2002) 945-969. | Zbl

[9] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). | MR | Zbl

[10] M. Dobrowolski, S. Gräf and C. Pflaum, On a posteriori error estimators in the finite element method on anisotropic meshes. Electron. Trans. Numer. Anal. 8 (1999) 36-45. | Zbl

[11] G. Kunert, A posteriori error estimation for anisotropic tetrahedral and triangular finite element meshes. Logos Verlag, Berlin (1999). Also Ph.D. thesis, TU Chemnitz, http://archiv.tu-chemnitz.de/pub/1999/0012/index.html | Zbl

[12] G. Kunert, An a posteriori residual error estimator for the finite element method on anisotropic tetrahedral meshes. Numer. Math. 86 (2000) 471-490, DOI 10.1007/s002110000170. | Zbl

[13] G. Kunert, A local problem error estimator for anisotropic tetrahedral finite element meshes. SIAM J. Numer. Anal. 39 (2001) 668-689. | Zbl

[14] G. Kunert, A posteriori L 2 error estimation on anisotropic tetrahedral finite element meshes. IMA J. Numer. Anal. 21 (2001) 503-523. | Zbl

[15] G. Kunert, Robust a posteriori error estimation for a singularly perturbed reaction-diffusion equation on anisotropic tetrahedral meshes. Adv. Comput. Math. 15 (2001) 237-259. | Zbl

[16] G. Kunert and S. Nicaise, Zienkiewicz-Zhu error estimators on anisotropic tetrahedral and triangular finite element meshes, preprint SFB393/01-20, TU Chemnitz, July 2001. Also http://archiv.tu-chemnitz.de/pub/2001/0059/index.html

[17] G. Kunert and R. Verfürth, Edge residuals dominate a posteriori error estimates for linear finite element methods on anisotropic triangular and tetrahedral meshes. Numer. Math. 86 (2000) 283-303, DOI 10.1007/s002110000152. | Zbl

[18] L.A. Oganesyan and L.A. Rukhovets, Variational-difference methods for the solution of elliptic equations. Izd. Akad. Nauk Armyanskoi SSR, Jerevan (1979), in Russian. | MR | Zbl

[19] G. Raugel, Résolution numérique par une méthode d'éléments finis du problème de Dirichlet pour le Laplacien dans un polygone. C. R. Acad. Sci. Paris, Sér. I Math 286 (1978) A791-A794. | Zbl

[20] R. Rodriguez, Some remarks on the Zienkiewicz-Zhu estimator. Numer. Meth. PDE 10 (1994) 625-635. | Zbl

[21] H.G. Roos and T. Linß, Gradient recovery for singularly perturbed boundary value problems II: Two-dimensional convection-diffusion. Math. Models Methods Appl. Sci. 11 (2001) 1169-1179. | Zbl

[22] K.G. Siebert, An a posteriori error estimator for anisotropic refinement. Numer. Math. 73 (1996) 373-398. | Zbl

[23] O. Steinbach, On the stability of the L 2 -projection in fractional Sobolev spaces. Numer. Math. 88 (2001) 367-379. | Zbl

[24] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, Chichester, Stuttgart (1996). | Zbl

[25] Zh. Zhang, Superconvergent finite element method on a Shishkin mesh for convection-diffusion problems. Report 98-006, Texas Tech University (1998).

[26] O.C. Zienkiewicz and J.Z. Zhu, A simple error estimator and adaptive procedure for practical engineering analysis. Internat. J. Numer. Methods Engrg. 24 (1987) 337-357. | Zbl

[27] O.C. Zienkiewicz and J.Z. Zhu, The superconvergent patch recovery (SPR) and adaptive finite element refinement. Comput. Methods Appl. Mech. Engrg. 101 (1992) 207-224. | Zbl

Cité par Sources :