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

Nicaise, Serge¹  ; Soualem, Nadir 

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 , Tome 39 (2005) no. 2, pp. 319-348.

Résumé

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.

Détail

DOI : 10.1051/m2an:2005009

Classification : 65N15, 65N30, 65M50
Mots-clés : error estimator, nonconforming FEM, heat equation

Bibliographie

[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. | Zbl

[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. | Zbl

[3] T. Apel, Anisotropic finite elements: Local estimates and applications. Adv. Numer. Math. Teubner, Stuttgart (1999). | Zbl | MR

[4] T. Apel and S. Nicaise, The inf-sup condition for some low order elements on anisotropic meshes. Calcolo 41 (2004) 89-113. | Zbl

[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. | Zbl

[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). | Zbl

[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. Finis 9 (2000) 425-438. | Zbl

[9] C. Bernardi and R. Verfürth, A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN 38 (2004) 437-455. | Zbl | mathdoc-id

[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. | mathdoc-id | Zbl | EuDML

[11] P. Ciarlet, The finite element method for elliptic problems. North Holland (1996). | Zbl | MR

[12] P. Clément, Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 2 (1975) 77-84. | mathdoc-id | Zbl | EuDML

[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. | Zbl

[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. | mathdoc-id | Zbl | EuDML

[15] V. Girault and P.-A. Raviart, Finite elements methods for Navier-Stokes equations, Theory and Algorithms. Springer Series in Computational Mathematics, Berlin (1986). | Zbl

[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. | Zbl

[17] M. Picasso, Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237. | Zbl

[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. | Zbl

[19] L.R. Scott and S. Zhang, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl

[20] R. Verfürth, A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, Chichester, Stuttgart (1996). | Zbl

[21] R. Verfürth, Error estimates for some quasi-interpolation operators. ESAIM: M2AN 33 (1999) 695-713. | mathdoc-id | Zbl | EuDML

[22] R. Verfürth, A posteriori error estimates for finite element discretization of the heat equation. Calcolo 40 (2003) 195-212. | Zbl