A posteriori error estimates for a nonconforming finite element discretization of the heat equation

Serge Nicaise; Nadir Soualem

ESAIM: Mathematical Modelling and Numerical Analysis (2010)

  • Volume: 39, Issue: 2, page 319-348
  • ISSN: 0764-583X

Abstract

top
The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in ℝ d , d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results with minimal assumptions on the mesh. Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.

How to cite

top

Nicaise, Serge, and Soualem, Nadir. " A posteriori error estimates for a nonconforming finite element discretization of the heat equation." ESAIM: Mathematical Modelling and Numerical Analysis 39.2 (2010): 319-348. <http://eudml.org/doc/194264>.

@article{Nicaise2010,
abstract = { The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in $\mathbb\{R\}^d$, d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results with minimal assumptions on the mesh. Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions. },
author = {Nicaise, Serge, Soualem, Nadir},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {Error estimator; nonconforming FEM; heat equation.; error estimator; nonconforming finite element; heat equation; backward Euler's scheme; numerical experiments; space-time adaptive algorithm},
language = {eng},
month = {3},
number = {2},
pages = {319-348},
publisher = {EDP Sciences},
title = { A posteriori error estimates for a nonconforming finite element discretization of the heat equation},
url = {http://eudml.org/doc/194264},
volume = {39},
year = {2010},
}

TY - JOUR
AU - Nicaise, Serge
AU - Soualem, Nadir
TI - A posteriori error estimates for a nonconforming finite element discretization of the heat equation
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2010/3//
PB - EDP Sciences
VL - 39
IS - 2
SP - 319
EP - 348
AB - The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in $\mathbb{R}^d$, d=2 or 3, using backward Euler's scheme. For this discretization, we derive a residual indicator, which use a spatial residual indicator based on the jumps of normal and tangential derivatives of the nonconforming approximation and a time residual indicator based on the jump of broken gradients at each time step. Lower and upper bounds form the main results with minimal assumptions on the mesh. Numerical experiments and a space-time adaptive algorithm confirm the theoretical predictions.
LA - eng
KW - Error estimator; nonconforming FEM; heat equation.; error estimator; nonconforming finite element; heat equation; backward Euler's scheme; numerical experiments; space-time adaptive algorithm
UR - http://eudml.org/doc/194264
ER -

References

top
  1. Y. Achdou, C. Bernardi and F. Coquel, A priori and a posteriori error analysis of finite volume discretizations of Darcy's equations. Numer. Math.96 (2003) 17–42.  
  2. G. Acosta and R.G. Durán, The maximum angle condition for mixed and non-conforming elements, Application to the Stokes equations. SIAM J. Numer. Anal.37 (1999) 18–36.  
  3. T. Apel, Anisotropic finite elements: Local estimates and applications. Adv. Numer. Math. Teubner, Stuttgart (1999).  
  4. T. Apel and S. Nicaise, The inf-sup condition for some low order elements on anisotropic meshes. Calcolo41 (2004) 89–113.  
  5. T. Apel, S. Nicaise and J. Schröberl, A non-conforming finite element method with anisotropic mesh grading for the stokes problem in domains with edges. IMA J. Numer. Anal.21 (2001) 843–856.  
  6. A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of some parabolic problem. Preprint Laboratoire J.-L. Lions 01045, Université Paris 6 (2001).  
  7. A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of a nonlinear parabolic equation. (2004) (to appear).  
  8. C. Bernardi and B. Métivet, Indicateurs d'erreur pour l'équation de la chaleur. Rev. Européenne Élém. Finis9 (2000) 425–438.  
  9. C. Bernardi and R. Verfürth, A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN38 (2004) 437–455.  
  10. P. Brenner, M. Crouzeix and V. Thomée, Single step methods for inhomogeneous linear differential equations in banach space. RAIRO Anal. Numér.16 (1982) 5–26.  
  11. P. Ciarlet, The finite element method for elliptic problems. North Holland (1996).  
  12. P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér.2 (1975) 77–84.  
  13. E. Creusé, G. Kunert and S. Nicaise, A posteriori error estimation for the Stokes problem: Anisotropic and isotropic discretizations. Math. Models Methods Appl. Sci.14 (2004) 1297–1341.  
  14. 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.  
  15. V. Girault and P.-A. Raviart, Finite elements methods for Navier-Stokes equations, Theory and Algorithms. Springer Series in Computational Mathematics, Berlin (1986).  
  16. C. Johnson, Y.-Y. Nie and V. Thomée, An a posteriori error estimate and adaptive timestep control for a backward Euler discretization of a parabolic problem. SIAM J. Numer. Anal.27 (1990) 277–291.  
  17. M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg.167 (1998) 223–237.  
  18. M. Picasso, An anisotropic error indicator based on Zienkiewicz-Zhu error estimator: Application to elliptic and parabolic problems. SIAM J. Sci. Comput.24 (2003) 1328–1355.  
  19. L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp.54 (1990) 483–493.  
  20. R. Verfürth, A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley-Teubner, Chichester, Stuttgart (1996).  
  21. R. Verfürth, Error estimates for some quasi-interpolation operators. ESAIM: M2AN33 (1999) 695–713.  
  22. R. Verfürth, A posteriori error estimates for finite element discretization of the heat equation. Calcolo40 (2003) 195–212.  

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.