The paper presents an a posteriori error estimator for a (piecewise linear) nonconforming finite element approximation of the heat equation in , 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.
Mots-clés : error estimator, nonconforming FEM, heat equation
@article{M2AN_2005__39_2_319_0,
author = {Nicaise, Serge and Soualem, Nadir},
title = {A posteriori error estimates for a nonconforming finite element discretization of the heat equation},
journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
pages = {319--348},
publisher = {EDP-Sciences},
volume = {39},
number = {2},
year = {2005},
doi = {10.1051/m2an:2005009},
mrnumber = {2143951},
zbl = {1078.65079},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an:2005009/}
}
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: Modélisation mathématique et analyse numérique PY - 2005 SP - 319 EP - 348 VL - 39 IS - 2 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an:2005009/ DO - 10.1051/m2an:2005009 LA - en ID - M2AN_2005__39_2_319_0 ER -
%0 Journal Article %A Nicaise, Serge %A Soualem, Nadir %T A posteriori error estimates for a nonconforming finite element discretization of the heat equation %J ESAIM: Modélisation mathématique et analyse numérique %D 2005 %P 319-348 %V 39 %N 2 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an:2005009/ %R 10.1051/m2an:2005009 %G en %F M2AN_2005__39_2_319_0
Nicaise, Serge; Soualem, Nadir. A posteriori error estimates for a nonconforming finite element discretization of the heat equation. ESAIM: Modélisation mathématique et analyse numérique, Tome 39 (2005) no. 2, pp. 319-348. doi : 10.1051/m2an:2005009. http://www.numdam.org/articles/10.1051/m2an:2005009/
[1] , and, A priori and a posteriori error analysis of finite volume discretizations of Darcy's equations. Numer. Math. 96 (2003) 17-42. | Zbl
[2] and, 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] , Anisotropic finite elements: Local estimates and applications. Adv. Numer. Math. Teubner, Stuttgart (1999). | MR | Zbl
[4] and, The inf-sup condition for some low order elements on anisotropic meshes. Calcolo 41 (2004) 89-113. | Zbl
[5] , and, 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] , and, A posteriori analysis of the finite element discretization of some parabolic problem. Preprint Laboratoire J.-L. Lions 01045, Université Paris 6 (2001). | Zbl
[7] , and, A posteriori analysis of the finite element discretization of a nonlinear parabolic equation. (2004) (to appear).
[8] and, Indicateurs d'erreur pour l'équation de la chaleur. Rev. Européenne Élém. Finis 9 (2000) 425-438. | Zbl
[9] and, A posteriori error analysis of the fully discretized time-dependent Stokes equations. ESAIM: M2AN 38 (2004) 437-455. | Numdam | Zbl
[10] , and, Single step methods for inhomogeneous linear differential equations in banach space. RAIRO Anal. Numér. 16 (1982) 5-26. | EuDML | Numdam | Zbl
[11] , The finite element method for elliptic problems. North Holland (1996). | MR | Zbl
[12] , Approximation by finite element functions using local regularization. RAIRO Anal. Numér. 2 (1975) 77-84. | EuDML | Numdam | Zbl
[13] , and, A posteriori error estimation for the Stokes problem: Anisotropic and isotropic discretizations. Math. Models Methods Appl. Sci. 14 (2004) 1297-1341. | Zbl
[14] ,, and, A posteriori error estimators for nonconforming finite element methods. RAIRO Modél. Math. Anal. Numér. 30 (1996) 385-400. | EuDML | Numdam | Zbl
[15] and, Finite elements methods for Navier-Stokes equations, Theory and Algorithms. Springer Series in Computational Mathematics, Berlin (1986). | Zbl
[16] , and, 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] , Adaptive finite elements for a linear parabolic problem. Comput. Methods Appl. Mech. Engrg. 167 (1998) 223-237. | Zbl
[18] , 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] and, Finite element interpolation of non-smooth functions satisfying boundary conditions. Math. Comp. 54 (1990) 483-493. | Zbl
[20] , A review of a posteriori error estimation and adaptive mesh-refinement techniques. Wiley-Teubner, Chichester, Stuttgart (1996). | Zbl
[21] , Error estimates for some quasi-interpolation operators. ESAIM: M2AN 33 (1999) 695-713. | EuDML | Numdam | Zbl
[22] , A posteriori error estimates for finite element discretization of the heat equation. Calcolo 40 (2003) 195-212. | Zbl
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