Nous présentons une méthode dʼéléments finis de type relaxation de temps pour les écoulements de fluides à grand nombre de Reynolds. Cette approche utilise des projections locales sur un espace de polynômes défini sur des macor-éléments pour chaque paire dʼéléments adjacents à une face intérieure du maillage. Nous démontrons la stabilité et la convergence. Nous donnons des résultats pour lʼéquation de convection–diffusion et les équations instationnaires dʼEuler. Cette note se termine par des résultats numériques.
We discuss a finite element time-relaxation method for high Reynolds number flows. The method uses local projections on polynomials defined on macroelements of each pair of two elements sharing a face. We prove that this method shares the optimal stability and convergence properties of the continuous interior penalty (CIP) method. We give the formulation both for the scalar convection–diffusion equation and the time-dependent incompressible Euler equations and the associated convergence results. This note finishes with some numerical illustrations.
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@article{CRMATH_2011__349_5-6_353_0,
author = {Becker, Roland and Burman, Erik and Hansbo, Peter},
title = {A finite element time relaxation method},
journal = {Comptes Rendus. Math\'ematique},
pages = {353--356},
publisher = {Elsevier},
volume = {349},
number = {5-6},
year = {2011},
doi = {10.1016/j.crma.2010.12.010},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2010.12.010/}
}
TY - JOUR AU - Becker, Roland AU - Burman, Erik AU - Hansbo, Peter TI - A finite element time relaxation method JO - Comptes Rendus. Mathématique PY - 2011 SP - 353 EP - 356 VL - 349 IS - 5-6 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2010.12.010/ DO - 10.1016/j.crma.2010.12.010 LA - en ID - CRMATH_2011__349_5-6_353_0 ER -
%0 Journal Article %A Becker, Roland %A Burman, Erik %A Hansbo, Peter %T A finite element time relaxation method %J Comptes Rendus. Mathématique %D 2011 %P 353-356 %V 349 %N 5-6 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2010.12.010/ %R 10.1016/j.crma.2010.12.010 %G en %F CRMATH_2011__349_5-6_353_0
Becker, Roland; Burman, Erik; Hansbo, Peter. A finite element time relaxation method. Comptes Rendus. Mathématique, Tome 349 (2011) no. 5-6, pp. 353-356. doi : 10.1016/j.crma.2010.12.010. http://www.numdam.org/articles/10.1016/j.crma.2010.12.010/
[1] A subgrid-scale deconvolution approach for shock capturing, J. Comput. Phys., Volume 178 (2002), pp. 391-426
[2] A two-level stabilization scheme for the Navier–Stokes equations, Numerical Mathematics and Advanced Applications, Springer, Berlin, 2004, pp. 123-130
[3] Interior penalty variational multiscale method for the incompressible Navier–Stokes equation: Monitoring artificial dissipation, Comput. Methods Appl. Mech. Engrg., Volume 196 (2007), pp. 4045-4058
[4] Continuous interior penalty finite element method for the time-dependent Navier–Stokes equations: Space discretization and convergence, Numer. Math., Volume 107 (2007), pp. 39-77
[5] Weighted error estimates of the continuous interior penalty method for singularly perturbed problems, IMA J. Numer. Anal., Volume 29 (2009), pp. 284-314
[6] Edge stabilization for Galerkin approximations of convection–diffusion–reaction problems, Comput. Methods Appl. Mech. Engrg., Volume 193 (2004), pp. 1437-1453
[7] On the accuracy of the finite element method plus time relaxation, Math. Comp., Volume 79 (2010), pp. 619-648
[8] Stabilization of Galerkin approximations of transport equations by subgrid modeling, Modél. Math. Anal. Numér., Volume 33 (1999), pp. 1293-1316
[9] A velocity–pressure streamline diffusion finite element method for the incompressible Navier–Stokes equations, Comput. Methods Appl. Mech. Engrg., Volume 84 (1990), pp. 175-192
[10] The dissipative structure of variational multiscale methods for incompressible flows, Comput. Methods Appl. Mech. Engrg., Volume 199 (2010), pp. 791-801
[11] Some historical remarks on the use of nonlinear viscosities, Large Eddy Simulation of Complex Engineering and Geophysical Flows, Cambridge Univ. Press, New York, 1993, pp. 3-36
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