Two-sided bounds of the discretization error for finite elements

Michal Křížek; Hans-Goerg Roos; Wei Chen

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2011)

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

Abstract

top
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.

How to cite

top

Křížek, 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. [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. [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. [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. [4] P.G. Ciarlet, The finite element method for elliptic problems. North-Holland, Amsterdam (1978). Zbl0511.65078MR520174
  5. [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. [6] Ch. Grossmann, H.-G. Roos and M. Stynes, Numerical treatment of partial differential equations. Springer-Verlag, Berlin, Heidelberg (2007). Zbl1180.65147MR2362757
  7. [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. [8] M. Křížek and P. Neittaanmäki, Finite element approximation of variational problems and applications. Longman Scientific & Technical, Harlow (1990). Zbl0708.65106
  9. [9] M. Křížek and P. Neittaanmäki, Mathematical and Numerical Modelling in Electrical Engineering: Theory and Applications. Kluwer, Dordrecht (1996). Zbl0859.65128MR1431889
  10. [10] Q. Lin and J. Lin, Finite element methods: Accuracy and improvement. Science Press, Beijing (2006). 
  11. [11] G.I. Marchuk and V.I. Agoshkov, Introduction aux méthodes des éléments finis. Mir, Moscow (1985). Zbl0645.65073
  12. [12] J. Nečas and I. Hlaváček, Mathematical theory of elastic and elasto-plastic bodies: An introduction. Elsevier, Amsterdam (1981). Zbl0448.73009
  13. [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. [14] G. Strang and G. Fix, An analysis of the finite element method. Prentice-Hall, Inc., Englewood Cliffs, New Jersey (1973). Zbl0356.65096MR443377
  15. [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. [16] L.B. Wahlbin, Superconvergence in Galerkin finite element methods, Lect. Notes in Math. 1605. Springer, Berlin (1995). Zbl0826.65092MR1439050
  17. [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. [18] N.N. Yan, Superconvergence analysis and a posteriori error estimation in finite element methods. Science Press, Beijing (2008). 

NotesEmbed ?

top

You must be logged in to post comments.

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

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.