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Two-sided bounds of the discretization error for finite elements

Published online by Cambridge University Press:  06 April 2011

Michal Křížek ,
Hans-Goerg Roos  and
Wei Chen
Michal Křížek
Affiliation:
Institute of Mathematics, Academy of Sciences, Žitná 25, 115 67 Prague 1, Czech Republic. krizek@math.cas.cz
Hans-Goerg Roos
Affiliation:
Institute of Numerical Mathematics, Technical University Dresden, Zellescher Weg 12–14, 01069 Dresden, Germany. hans-goerg.roos@tu-dresden.de
Wei Chen
Affiliation:
School of Economics, Shandong University, 27 Shanda Nanlu, Jinan 250 100, P.R. China. weichen@sdu.edu.cn
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Abstract

We derive an optimal lower bound of theinterpolation error for linear finite elements on a bounded two-dimensionaldomain. Using the supercloseness between the linear interpolantof the true solution of an elliptic problem and its finite elementsolution on uniform partitions, we furtherobtain two-sided a priori bounds of the discretization error by means of theinterpolation error. Two-sided bounds for bilinear finite elementsare given as well. Numerical tests illustrate our theoreticalanalysis.

Keywords

Lagrange finite elementsCéa's lemmasuperconvergencelower error estimates.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 45 , Issue 5 , September 2011 , pp. 915 - 924
Copyright
© EDP Sciences, SMAI, 2011

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References

Brandts, J. and Křížek, M., Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal. 23 (2003) 489–505. CrossRef
Chen, W. and Křížek, M., What is the smallest possible constant in Céa's lemma? Appl. Math. 51 (2006) 128–144.
Chen, W. and Křížek, M., Lower bounds for the interpolation error for finite elements. Mathematics in Practice and Theory 39 (2009) 159–164 (in Chinese).
P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978).
Franz, S. and Linss, T., Superconvergence analysis of the Galerkin FEM for a singularly perturbed convection-diffusion problems with characteristic layers. Numer. Methods Partial Differ. Equ. 24 (2008) 144–164. CrossRef
Ch. Grossmann, H.-G. Roos and M. Stynes, Numerical treatment of partial differential equations. Springer-Verlag, Berlin, Heidelberg (2007).
Korotov, S., Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions. Appl. Math. 52 (2007) 235–249. CrossRef
M. Křížek and P. Neittaanmäki, Finite element approximation of variational problems and applications. Longman Scientific & Technical, Harlow (1990).
M. Křížek and P. Neittaanmäki, Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer, Dordrecht (1996).
Q. Lin and J. Lin, Finite element methods: Accuracy and improvement. Science Press, Beijing (2006).
G.I. Marchuk and V.I. Agoshkov, Introduction aux méthodes des éléments finis. Mir, Moscow (1985).
J. Nečas and I. Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier, Amsterdam (1981).
Oganesjan, L.A. and Ruhovec, L.A., An investigation of the rate of convergence of variational-difference schemes for second order elliptic equations in a two-dimensional region with smooth boundary. Ž. Vyčisl. Mat. i Mat. Fyz. 9 (1969) 1102–1120.
G. Strang and G. Fix, An analysis of the finite element method. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1973).
R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. John Wiley & Sons, Chichester, Teubner, Stuttgart (1996).
L.B. Wahlbin, Superconvergence in Galerkin finite element methods, Lect. Notes in Math. 1605. Springer, Berlin (1995).
Xu, L. and Zhang, Z., Analysis of recovery type a posteriori error estimation for mildly structured grids. Math. Comp. 73 (2004) 1139–1152. CrossRef
N.N. Yan, Superconvergence analysis and a posteriori error estimation in finite element methods. Science Press, Beijing (2008).