# Two-sided bounds of the discretization error for finite elements

Michal Křížek; Hans-Goerg Roos; Wei Chen

- Volume: 45, Issue: 5, page 915-924
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topKřížek, Michal, Roos, Hans-Goerg, and Chen, Wei. "Two-sided bounds of the discretization error for finite elements." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 45.5 (2011): 915-924. <http://eudml.org/doc/273106>.

@article{Křížek2011,

abstract = {We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided a priori bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis.},

author = {Křížek, Michal, Roos, Hans-Goerg, Chen, Wei},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},

keywords = {Lagrange finite elements; Céa's lemma; superconvergence; lower error estimates; Céa’s lemma; numerical examples; elliptic problem},

language = {eng},

number = {5},

pages = {915-924},

publisher = {EDP-Sciences},

title = {Two-sided bounds of the discretization error for finite elements},

url = {http://eudml.org/doc/273106},

volume = {45},

year = {2011},

}

TY - JOUR

AU - Křížek, Michal

AU - Roos, Hans-Goerg

AU - Chen, Wei

TI - Two-sided bounds of the discretization error for finite elements

JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

PY - 2011

PB - EDP-Sciences

VL - 45

IS - 5

SP - 915

EP - 924

AB - We derive an optimal lower bound of the interpolation error for linear finite elements on a bounded two-dimensional domain. Using the supercloseness between the linear interpolant of the true solution of an elliptic problem and its finite element solution on uniform partitions, we further obtain two-sided a priori bounds of the discretization error by means of the interpolation error. Two-sided bounds for bilinear finite elements are given as well. Numerical tests illustrate our theoretical analysis.

LA - eng

KW - Lagrange finite elements; Céa's lemma; superconvergence; lower error estimates; Céa’s lemma; numerical examples; elliptic problem

UR - http://eudml.org/doc/273106

ER -

## References

top- [1] J. Brandts and M. Křížek, Gradient superconvergence on uniform simplicial partitions of polytopes. IMA J. Numer. Anal.23 (2003) 489–505. Zbl1042.65081MR1987941
- [2] W. Chen and M. Křížek, What is the smallest possible constant in Céa's lemma? Appl. Math.51 (2006) 128–144. Zbl1164.65495
- [3] W. Chen and M. Křížek, Lower bounds for the interpolation error for finite elements. Mathematics in Practice and Theory 39 (2009) 159–164 (in Chinese). Zbl1212.41001MR2599063
- [4] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). Zbl0511.65078MR520174
- [5] S. Franz and T. Linss, 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. Zbl1133.65090MR2371352
- [6] Ch. Grossmann, H.-G. Roos and M. Stynes, Numerical treatment of partial differential equations. Springer-Verlag, Berlin, Heidelberg (2007). Zbl1180.65147MR2362757
- [7] S. Korotov, Two-sided a posteriori error estimates for linear elliptic problems with mixed boundary conditions. Appl. Math.52 (2007) 235–249. Zbl1164.65485MR2316154
- [8] M. Křížek and P. Neittaanmäki, Finite element approximation of variational problems and applications. Longman Scientific & Technical, Harlow (1990). Zbl0708.65106
- [9] M. Křížek and P. Neittaanmäki, Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer, Dordrecht (1996). Zbl0859.65128MR1431889
- [10] Q. Lin and J. Lin, Finite element methods: Accuracy and improvement. Science Press, Beijing (2006).
- [11] G.I. Marchuk and V.I. Agoshkov, Introduction aux méthodes des éléments finis. Mir, Moscow (1985). Zbl0645.65073
- [12] J. Nečas and I. Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier, Amsterdam (1981). Zbl0448.73009
- [13] L.A. Oganesjan and L.A. Ruhovec, 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. Zbl0234.65093MR295599
- [14] G. Strang and G. Fix, An analysis of the finite element method. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1973). Zbl0356.65096MR443377
- [15] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. John Wiley & Sons, Chichester, Teubner, Stuttgart (1996). Zbl0853.65108
- [16] L.B. Wahlbin, Superconvergence in Galerkin finite element methods, Lect. Notes in Math. 1605. Springer, Berlin (1995). Zbl0826.65092MR1439050
- [17] L. Xu and Z. Zhang, Analysis of recovery type a posteriori error estimation for mildly structured grids. Math. Comp.73 (2004) 1139–1152. Zbl1050.65103MR2047081
- [18] N.N. Yan, Superconvergence analysis and a posteriori error estimation in finite element methods. Science Press, Beijing (2008).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.