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

Serge Nicaise[1]; Nadir Soualem

  • [1] Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique (2005)

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

Abstract

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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

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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 - Modélisation Mathématique et Analyse Numérique 39.2 (2005): 319-348. <http://eudml.org/doc/245732>.

@article{Nicaise2005,
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.},
affiliation = {Université de Valenciennes et du Hainaut Cambrésis, MACS, Le Mont Houy, 59313 Valenciennes Cedex 9, France. http://www.univ-valenciennes.fr/macs/Serge.Nicaise},
author = {Nicaise, Serge, Soualem, Nadir},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique},
keywords = {error estimator; nonconforming FEM; heat equation; nonconforming finite element; backward Euler's scheme; numerical experiments; space-time adaptive algorithm},
language = {eng},
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/245732},
volume = {39},
year = {2005},
}

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 - Modélisation Mathématique et Analyse Numérique
PY - 2005
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; nonconforming finite element; backward Euler's scheme; numerical experiments; space-time adaptive algorithm
UR - http://eudml.org/doc/245732
ER -

References

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  1. [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. Zbl1050.76035
  2. [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. Zbl0948.65115
  3. [3] T. Apel, Anisotropic finite elements: Local estimates and applications. Adv. Numer. Math. Teubner, Stuttgart (1999). Zbl0934.65121MR1716824
  4. [4] T. Apel and S. Nicaise, The inf-sup condition for some low order elements on anisotropic meshes. Calcolo 41 (2004) 89–113. Zbl1108.65117
  5. [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. Zbl0998.65116
  6. [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). Zbl1072.65124
  7. [7] A. Bergam, C. Bernardi and Z. Mghazli, A posteriori analysis of the finite element discretization of a nonlinear parabolic equation. (2004) (to appear). Zbl1072.65124
  8. [8] C. Bernardi and B. Métivet, Indicateurs d’erreur pour l’équation de la chaleur. Rev. Européenne Élém. Finis 9 (2000) 425–438. Zbl0959.65106
  9. [9] C. Bernardi and R. Verfürth, A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN 38 (2004) 437–455. Zbl1079.76042
  10. [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. Zbl0477.65040
  11. [11] P. Ciarlet, The finite element method for elliptic problems. North Holland (1996). Zbl0383.65058MR520174
  12. [12] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 2 (1975) 77–84. Zbl0368.65008
  13. [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. Zbl1071.65142
  14. [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. Zbl0853.65110
  15. [15] V. Girault and P.-A. Raviart, Finite elements methods for Navier-Stokes equations, Theory and Algorithms. Springer Series in Computational Mathematics, Berlin (1986). Zbl0585.65077
  16. [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. Zbl0701.65063
  17. [17] M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223–237. Zbl0935.65105
  18. [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. Zbl1061.65116
  19. [19] L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483–493. Zbl0696.65007
  20. [20] R. Verfürth, A review of a posteriori error estimation and adaptive mesh–refinement techniques. Wiley-Teubner, Chichester, Stuttgart (1996). Zbl0853.65108
  21. [21] R. Verfürth, Error estimates for some quasi-interpolation operators. ESAIM: M2AN 33 (1999) 695–713. Zbl0938.65125
  22. [22] R. Verfürth, A posteriori error estimates for finite element discretization of the heat equation. Calcolo 40 (2003) 195–212. Zbl1168.65418

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