Numerical Analysis
Four closely related equilibrated flux reconstructions for nonconforming finite elements
Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 77-80.

We consider the Crouzeix–Raviart nonconforming finite element method for the Laplace equation. We present four equilibrated flux reconstructions, by direct prescription or by mixed approximation of local Neumann problems, either relying on the original simplicial mesh only or employing a dual mesh. We show that all these reconstructions coincide provided the underlying system of linear algebraic equations is solved exactly. We finally consider an inexact algebraic solve, adjust the flux reconstructions, and point out the differences.

Nous étudions la méthode des éléments finis non conformes de Crouzeix et Raviart pour lʼéquation de Laplace. Nous introduisons quatre reconstructions équilibrées du flux, par prescription directe ou par une approximation mixte de problèmes locaux de Neumann, soit sur le maillage simplectique de départ, soit sur un maillage dual. Nous montrons que toutes ces reconstructions coïncident si le système dʼéquations linéaires associé est résolu exactement. Nous considérons enfin une solution algébrique inexacte, ajustons les reconstructions du flux et indiquons les différences entre les reconstructions.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2013.01.001
Ern, Alexandre 1; Vohralík, Martin 2

1 Université Paris-Est, CERMICS, École des Ponts ParisTech, 77455 Marne-la-Vallée, France
2 INRIA Paris-Rocquencourt, B.P. 105, 78153 Le Chesnay, France
@article{CRMATH_2013__351_1-2_77_0,
     author = {Ern, Alexandre and Vohral{\'\i}k, Martin},
     title = {Four closely related equilibrated flux reconstructions for nonconforming finite elements},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {77--80},
     publisher = {Elsevier},
     volume = {351},
     number = {1-2},
     year = {2013},
     doi = {10.1016/j.crma.2013.01.001},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2013.01.001/}
}
TY  - JOUR
AU  - Ern, Alexandre
AU  - Vohralík, Martin
TI  - Four closely related equilibrated flux reconstructions for nonconforming finite elements
JO  - Comptes Rendus. Mathématique
PY  - 2013
SP  - 77
EP  - 80
VL  - 351
IS  - 1-2
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2013.01.001/
DO  - 10.1016/j.crma.2013.01.001
LA  - en
ID  - CRMATH_2013__351_1-2_77_0
ER  - 
%0 Journal Article
%A Ern, Alexandre
%A Vohralík, Martin
%T Four closely related equilibrated flux reconstructions for nonconforming finite elements
%J Comptes Rendus. Mathématique
%D 2013
%P 77-80
%V 351
%N 1-2
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2013.01.001/
%R 10.1016/j.crma.2013.01.001
%G en
%F CRMATH_2013__351_1-2_77_0
Ern, Alexandre; Vohralík, Martin. Four closely related equilibrated flux reconstructions for nonconforming finite elements. Comptes Rendus. Mathématique, Volume 351 (2013) no. 1-2, pp. 77-80. doi : 10.1016/j.crma.2013.01.001. http://www.numdam.org/articles/10.1016/j.crma.2013.01.001/

[1] Ainsworth, Mark Robust a posteriori error estimation for nonconforming finite element approximation, SIAM J. Numer. Anal., Volume 42 (2005) no. 6, pp. 2320-2341

[2] Braess, Dietrich An a posteriori error estimate and a comparison theorem for the nonconforming P1 element, Calcolo, Volume 46 (2009) no. 2, pp. 149-155

[3] Braess, Dietrich; Schöberl, Joachim Equilibrated residual error estimator for edge elements, Math. Comp., Volume 77 (2008) no. 262, pp. 651-672

[4] Destuynder, Philippe; Métivet, Brigitte Explicit error bounds for a nonconforming finite element method, SIAM J. Numer. Anal., Volume 35 (1998) no. 5, pp. 2099-2115

[5] Destuynder, Philippe; Métivet, Brigitte Explicit error bounds in a conforming finite element method, Math. Comp., Volume 68 (1999) no. 228, pp. 1379-1396

[6] Ern, Alexandre; Vohralík, Martin Flux reconstruction and a posteriori error estimation for discontinuous Galerkin methods on general nonmatching grids, C. R. Acad. Sci. Paris, Ser. I, Volume 347 (2009) no. 7–8, pp. 441-444

[7] Alexandre Ern, Martin Vohralík, Adaptive inexact Newton methods with a posteriori stopping criteria for nonlinear diffusion PDEs, HAL preprint 00681422 v2, submitted for publication, 2012.

[8] Hannukainen, Antti; Stenberg, Rolf; Vohralík, Martin A unified framework for a posteriori error estimation for the Stokes problem, Numer. Math., Volume 122 (2012) no. 4, pp. 725-769

[9] Jiránek, Pavel; Strakoš, Zdeněk; Vohralík, Martin A posteriori error estimates including algebraic error and stopping criteria for iterative solvers, SIAM J. Sci. Comput., Volume 32 (2010) no. 3, pp. 1567-1590

[10] Kim, Kwang Y. A posteriori error analysis for locally conservative mixed methods, Math. Comp., Volume 76 (2007) no. 257, pp. 43-66

[11] Luce, Robert; Wohlmuth, Barbara I. A local a posteriori error estimator based on equilibrated fluxes, SIAM J. Numer. Anal., Volume 42 (2004) no. 4, pp. 1394-1414

[12] Marini, L. Donatella An inexpensive method for the evaluation of the solution of the lowest order Raviart–Thomas mixed method, SIAM J. Numer. Anal., Volume 22 (1985) no. 3, pp. 493-496

[13] Vohralík, Martin Guaranteed and fully robust a posteriori error estimates for conforming discretizations of diffusion problems with discontinuous coefficients, J. Sci. Comput., Volume 46 (2011) no. 3, pp. 397-438

Cited by Sources:

This work was partly supported by the Groupement MoMaS (PACEN/CNRS, ANDRA, BRGM, CEA, EdF, IRSN) and by the ERT project “Enhanced oil recovery and geological sequestration of CO2: mesh adaptivity, a posteriori error control, and other advanced techniques” (LJLL UPMC/IFPEN).