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T-coercivity for solving Stokes problem with nonconforming finite elements

T-coercivité pour résoudre le problème de Stokes avec des éléments finis non conformes

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Abstract

We propose to analyse the discretization of the Stokes problem with nonconforming finite elements in light of the T-coercivity (cf. [1] for Helmholtz-like problems, see [2], [3] and [4] for the neutron diffusion equation). We propose explicit expressions of the stability constants. Finally, we give numerical results illustrating the importance of using divergence-free velocity reconstruction. [1] P. Ciarlet Jr. T-coercivity: Application to the discretization of Helmhotz-like problems. Computers & Mathematics with Applications, 64(1):22–24, 2012. [2] E. Jamelot and P. Ciarlet, Jr. Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation. Journal of Computational Physics, 241:445–463, 2013. [3] P. Ciarlet Jr., E. Jamelot, and F. D. Kpadonou. Domain decomposition methods for the diffusion equation with low-regularity solution. Computers & Mathematics with Applications, 74(10):2369–2384, 2017. [4] L. Giret. Non-Conforming Domain Decomposition for the Multigroup Neutron SPN Equation. PhD thesis, Universit´e Paris-Saclay, 2018.
Nous proposons d’analyser la discrétisation du problème de Stokes avec des éléments finis non conformes à la lumière de la T-coercitivité (cf. [1] pour les problèmes de type Helmholtz, voir [2], [3] et [4] pour l’équation de diffusion des neutrons). Enfin, nous donnons des résultats numériques illustrant l’importance d’utiliser une méthode de reconstruction de vitesse à divergence nulle. [1] P. Ciarlet Jr. T-coercivity: Application to the discretization of Helmhotz-like problems. Computers & Mathematics with Applications, 64(1):22–24, 2012. [2] E. Jamelot and P. Ciarlet, Jr. Fast non-overlapping Schwarz domain decomposition methods for solving the neutron diffusion equation. Journal of Computational Physics, 241:445–463, 2013. [3] P. Ciarlet Jr., E. Jamelot, and F. D. Kpadonou. Domain decomposition methods for the diffusion equation with low-regularity solution. Computers & Mathematics with Applications, 74(10):2369–2384, 2017. [4] L. Giret. Non-Conforming Domain Decomposition for the Multigroup Neutron SPN Equation. PhD thesis, Universit´e Paris-Saclay, 2018.
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cea-03833616 , version 1 (28-10-2022)

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Erell Jamelot. T-coercivity for solving Stokes problem with nonconforming finite elements. 2022. ⟨cea-03833616⟩
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