Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients

Stefano Berrone

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2006)

  • Volume: 40, Issue: 6, page 991-1021
  • ISSN: 0764-583X

Abstract

top
In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable θ -scheme with 1 / 2 ≤ θ ≤ 1 . Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313–348; Petzoldt, Adv. Comput. Math. 16 (2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.

How to cite

top

Berrone, Stefano. "Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients." ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique 40.6 (2006): 991-1021. <http://eudml.org/doc/244686>.

@article{Berrone2006,
abstract = {In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable $\theta $-scheme with $1/2\le \theta \le 1$. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313–348; Petzoldt, Adv. Comput. Math. 16 (2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.},
author = {Berrone, Stefano},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {a posteriori error estimates; parabolic problems; discontinuous coefficients; finite elements; heat equation},
language = {eng},
number = {6},
pages = {991-1021},
publisher = {EDP-Sciences},
title = {Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients},
url = {http://eudml.org/doc/244686},
volume = {40},
year = {2006},
}

TY - JOUR
AU - Berrone, Stefano
TI - Robust a posteriori error estimates for finite element discretizations of the heat equation with discontinuous coefficients
JO - ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique
PY - 2006
PB - EDP-Sciences
VL - 40
IS - 6
SP - 991
EP - 1021
AB - In this work we derive a posteriori error estimates based on equations residuals for the heat equation with discontinuous diffusivity coefficients. The estimates are based on a fully discrete scheme based on conforming finite elements in each time slab and on the A-stable $\theta $-scheme with $1/2\le \theta \le 1$. Following remarks of [Picasso, Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237; Verfürth, Calcolo 40 (2003) 195–212] it is easy to identify a time-discretization error-estimator and a space-discretization error-estimator. In this work we introduce a similar splitting for the data-approximation error in time and in space. Assuming the quasi-monotonicity condition [Dryja et al., Numer. Math. 72 (1996) 313–348; Petzoldt, Adv. Comput. Math. 16 (2002) 47–75] we have upper and lower bounds whose ratio is independent of any meshsize, timestep, problem parameter and its jumps.
LA - eng
KW - a posteriori error estimates; parabolic problems; discontinuous coefficients; finite elements; heat equation
UR - http://eudml.org/doc/244686
ER -

References

top
  1. [1] I. Babuška and W.C. Rheinboldt, Error estimates for adaptive finite element method. SIAM J. Numer. Anal. 15 (1978) 736–754. Zbl0398.65069
  2. [2] R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods. Acta Num. (2001) 1–102. Zbl1105.65349
  3. [3] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic equations. Math. Comp. 74 (2004) 1117–1138. Zbl1072.65124
  4. [4] C. Bernardi and R. Verfürth, Adaptive finite element methods for elliptic equations with non-smooth coefficients. Numer. Math. 85 (2000) 579–608. Zbl0962.65096
  5. [5] P.G. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978). Zbl0383.65058MR520174
  6. [6] P. Clément, Approximation by finite element functions using local regularization. RAIRO Sér. Rouge Anal. Numér. 9 (1975) 77–84. Zbl0368.65008
  7. [7] W. Dörfler, A convergent adaptive algorithm for Poisson’s equation. SIAM J. Numer. Anal. 33 (1996) 1106–1124. Zbl0854.65090
  8. [8] M. Dryja, M.V. Sarkis and O.B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions. Numer. Math. 72 (1996) 313–348. Zbl0857.65131
  9. [9] K. Eriksson and C. Johnson, Adaptive finite element methods for parabolic problems. V. Long-time integration. SIAM J. Numer. Anal. 32 (1995) 1750–1763. Zbl0835.65117
  10. [10] K. Eriksson, D. Estep, P. Hansbo and C. Johnson, Introduction to adaptive methods for differential equations. Acta Num. (1995) 105–158. Zbl0829.65122
  11. [11] B.S. Kirk, J.W. Peterson, R. Stogner and S. Petersen, LibMesh. The University of Texas, Austin, CFDLab and Technische Universität Hamburg, Hamburg. http://libmesh.sourceforge.net. 
  12. [12] P. Morin, R.H. Nocetto and K.G. Siebert, Convergence of adaptive finite element methods. SIAM Rev. 44 (2002) 631–658. Zbl1016.65074
  13. [13] M. Petzoldt, A posteriori error estimators for elliptic equations with discontinuous coefficients. Adv. Comput. Math. 16 (2002) 47–75. Zbl0997.65123
  14. [14] M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237. Zbl0935.65105
  15. [15] R. Verfürth, A posteriori error estimates for nonlinear problems. Finite element discretizations of parabolic equations. Ruhr-Universität Bochum, Report 180/1995. Zbl0974.65087
  16. [16] R. Verfürth, A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques. John Wiley & Sons, Chichester-New York (1996). Zbl0853.65108
  17. [17] R. Verfürth, A posteriori error estimates for finite element discretization of the heat equations. Calcolo 40 (2003) 195–212. Zbl1168.65418
  18. [18] 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. Zbl0602.73063

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.