A posteriori error estimates with post-processing for nonconforming finite elements
Schieweck, Friedhelm
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002), p. 489-503 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

For a nonconforming finite element approximation of an elliptic model problem, we propose a posteriori error estimates in the energy norm which use as an additive term the “post-processing error” between the original nonconforming finite element solution and an easy computable conforming approximation of that solution. Thus, for the error analysis, the existing theory from the conforming case can be used together with some simple additional arguments. As an essential point, the property is exploited that the nonconforming finite element space contains as a subspace a conforming finite element space of first order. This property is fulfilled for many known nonconforming spaces. We prove local lower and global upper a posteriori error estimates for an enhanced error measure which is the discretization error in the discrete energy norm plus the error of the best representation of the exact solution by a function in the conforming space used for the post-processing. We demonstrate that the idea to use a computed conforming approximation of the nonconforming solution can be applied also to derive an a posteriori error estimate for a linear functional of the solution which represents some quantity of physical interest.

Publié le : 2002-01-01
DOI : https://doi.org/10.1051/m2an:2002022
Classification:  65N15,  65N30 
@article{M2AN_2002__36_3_489_0,
     author = {Schieweck, Friedhelm},
     title = {A posteriori error estimates with post-processing for nonconforming finite elements},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {36},
     year = {2002},
     pages = {489-503},
     doi = {10.1051/m2an:2002022},
     mrnumber = {1918941},
     zbl = {1041.65083},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2002__36_3_489_0}
}

Schieweck, Friedhelm. A posteriori error estimates with post-processing for nonconforming finite elements. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 36 (2002) pp. 489-503. doi : 10.1051/m2an:2002022. http://gdmltest.u-ga.fr/item/M2AN_2002__36_3_489_0/

[1] M. Ainsworth and J.T. Oden, A posteriori error estimation in finite element analysis. Comput. Methods Appl. Mech. Engrg. 142 (1997) 1-88. | Zbl 0895.76040

[2] L. Angermann, A posteriori error estimates for FEM with violated Galerkin orthogonality. Preprint 27/98, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik (1998). | MR 1902295 | Zbl 1003.65058

[3] R. Becker, M. Braack, R. Rannacher and C. Waguet, Fast and reliable solution of the Navier-Stokes equations including chemistry. Comput. Vis. Sci. 2 (1999) 107-122. | Zbl 1067.76560

[4] R. Becker and R. Rannacher, A feed-back approach to error control in finite element methods: Basic analysis and examples. East-West J. Numer. Math. 4 (1996) 237-264. | Zbl 0868.65076

[5] C. Bernardi and V. Girault, A local regularization operator for triangular and quadrilateral finite elements. SIAM J. Numer. Anal. 35 (1998) 1893-1916. | Zbl 0913.65007

[6] F. Brezzi and M. Fortin, Mixed and Hybrid Finite Element Methods. Vol. 15 of Springer Ser. Comput. Math. Springer-Verlag (1991). | MR 1115205 | Zbl 0788.73002

[7] Z. Cai, Jr. J. Douglas and X. Ye, A stable nonconforming quadrilateral finite element method for the stationary Stokes and Navier-Stokes equations. Calcolo 36 (1999) 215-232. | Zbl 0947.76047

[8] C. Carstensen, S. Bartels and S. Jansche, A posteriori error estimates for nonconforming finite element methods. Berichtsreihe des Mathematischen Seminars Kiel, Report Nr. 00-13, Christian-Albrechts-Universität zu Kiel (2000). | Zbl 1010.65044

[9] M. Crouzeix and P.-A. Raviart, Conforming and nonconforming finite element methods for solving the stationary Stokes equations I. RAIRO Anal. Numér. 7 (1973) 33-76. | Numdam | Zbl 0302.65087

[10] E. Dari, R. Durán and C. Padra, Error estimators for nonconforming finite element approximations of the Stokes problem. Math. Comp. 64 (1995) 1017-1033. | Zbl 0827.76042

[11] E. Dari, R. Durán, C. Padra and V. Vampa, A posteriori error estimators for nonconforming finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 385-400. | Numdam | Zbl 0853.65110

[12] V. Girault and P.-A. Raviart, Finite Element Methods for Navier-Stokes equations. Springer-Verlag, Berlin, Heidelberg, New York (1986). | Zbl 0585.65077

[13] J.-P. Hennart, J. Jaffre and J.E. Roberts, A constructive method for deriving finite elements of nodal type. Numer. Math. 53 (1988) 701-738. | Zbl 0677.65101

[14] R.H.W. Hoppe and B. Wohlmuth, Element-oriented and edge-oriented local error estimators for nonconforming finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 237-263. | Numdam | Zbl 0843.65075

[15] V. John, A posteriori error estimators for the nonconforming P1-finite element discretization of convection-diffusion equations. Preprint 10/97, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik (1997). http://www-ian.math.uni-magdeburg.de/home/john/.

[16] V. John, A posteriori L 2 -error estimates for the nonconforming P 1 /P 0 -finite element discretization of the Stokes equations. J. Comput. Appl. Math. 96 (1998) 99-116. | Zbl 0930.65123

[17] G. Kanschat and F.-T. Suttmeier, A posteriori error estimates for nonconforming finite element schemes. Calcolo 36 (1999) 129-141. | Zbl 0936.65128

[18] R. Rannacher, Error control in finite element computations. Preprint 98-54, Universität Heidelberg, IWR (1998). http://www.iwr.uni-heidelberg.de/.

[19] R. Rannacher, Adaptive Galerkin finite element methods for partial differential equations. J. Comput. Appl. Math. 128 (2001) 205-233. | Zbl 0976.65101

[20] R. Rannacher and S. Turek, Simple nonconforming quadrilateral stokes element. Numer. Methods Partial Differential Equations 8 (1992) 97-111. | Zbl 0742.76051

[21] F. Schieweck, A parallel multigrid algorithm for solving the Navier-Stokes equations. IMPACT Comput. Sci. Eng. 5 (1993) 345-378. | Zbl 0789.76065

[22] F. Schieweck, Parallele Lösung der stationären inkompressiblen Navier-Stokes Gleichungen. Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik (1996). Habilitation. http://www-ian.math.uni-magdeburg.de/home/schieweck. | Zbl 0915.76051

[23] F. Schieweck, A general transfer operator for arbitrary finite element spaces. Preprint 25/00, Otto-von-Guericke Universität Magdeburg, Fakultät für Mathematik (2000). http://www-ian.math.uni-magdeburg.de/home/schieweck.

[24] L.R. Scott and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl 0696.65007

[25] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. Wiley-Teubner series in advances in numerical mathematics, Wiley-Teubner (1996). | Zbl 0853.65108