T-coercivity for solving Stokes problem with nonconforming finite elements

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

Schwarz type preconditioners for the neutron diffusion equation

[EN] Domain decomposition is a mature methodology that has been used to accelerate the convergence of partial differential equations. Even if it was devised as a solver by itself, it is usually employed together with Krylov iterative methods improving its rate of convergence, and providing scalability with respect to the size of the problem. In this work, a high order finite element discretization of the neutron diffusion equation is considered. In this problem the preconditioning of large and sparse linear systems arising from a source driven formulation becomes necessary due to the complexity of the problem. On the other hand, preconditioners based on an incomplete factorization are very expensive from the point of view of memory requirements. The acceleration of the neutron diffusion equation is thus studied here by using alternative preconditioners based on domain decomposition techniques inside Schur complement methodology. The study considers substructuring preconditioners, which do not involve overlapping, and additive Schwarz preconditioners, where some overlapping between the subdomains is taken into account. The performance of the different approaches is studied numerically using two-dimensional and three-dimensional problems. It is shown that some of the proposed methodologies outperform incomplete LU factorization for preconditioning as long as the linear system to be solved is large enough, as it occurs for three-dimensional problems. They also outperform classical diagonal Jacobi preconditioners, as long as the number of systems to be solved is large enough in such a way that the overhead of building the pre-conditioner is less than the improvement in the convergence rate. (C) 2016 Elsevier B.V. All rights reserved.The work has been partially supported by the spanish Ministerio de Economía y Competitividad under projects ENE 2014-59442-P and MTM2014-58159-P, the Generalitat Valenciana under the project PROMETEO II/2014/008 and the Universitat Politècnica de València under the project FPI-2013. The work has also been supported partially by the Swedish Research Council (VR-Vetenskapsrådet) within a framework grant called DREAM4SAFER, research contract C0467701.Vidal-Ferràndiz, A.; González Pintor, S.; Ginestar Peiro, D.; Verdú Martín, GJ.; Demazière, C. (2017). Schwarz type preconditioners for the neutron diffusion equation. Journal of Computational and Applied Mathematics. 309:563-574. https://doi.org/10.1016/j.cam.2016.02.056S56357430

Nodal finite element methods for Maxwell's equations [Eléments finis nodaux pour les équations de Maxwell]

International audienceAn original approach of the singular complement method for Maxwell's equations in bounded polygonal domains is presented. A splitting of the electric field à la Moussaoui is proposed: E=ER+λxP, where ER∈H1(ω)², λ depends on the data and domain and xP is known explicitly. The same splitting can used for the magnetic field. No cut-off function is needed and improved error estimates are derived. © 2004 Académie des sciences. Publié par Elsevier SAS. Tous droits réservés

Estimations de stabilité pour résoudre le problème de Stokes avec des éléments finis non conformes.

We propose to analyse the discretization of the Stokes problem with nonconforming finite elements in light of the T-coercivity. First we exhibit a family of operators to prove T-coercivity and we show that the stability constant is equal to the classical one up to a constant which depends on the Babuska-Aziz constant. Then we explicit the stability constants with respect to the shape regularity parameter for order 1 in 2 or 3 dimension, and order 2 in 2 dimension. In this last case, we improve the result of the original Fortin-Soulie paper. Second, we illustrate the importance of using a divergence-free velocity reconstruction on some numerical experiments.Nous proposons d'analyser la discrétisation du problème de Stokes avec des éléments finis non conformes à la lumière de la T-coercivité. Tout d'abord, pour prouver la T-coercitivité, nous exhibons une famille d'opérateurs et nous montrons que la constante de stabilité est égale à la constante de stabilité classique, à une constante près qui dépend de la constante de Babuska-Aziz. Par la suite, nous explicitons les constantes de stabilité par rapport au paramètre de régularité de forme pour l'ordre 1 en dimension 2 ou 3, et l'ordre 2 en dimension 2. Dans ce dernier cas, nous améliorons le résultat de l'article original de Fortin-Soulie. Ensuite nous illustrons l'importance d'utiliser une méthode de projection conforme dans H(div) pour certaines expériences numériques

Constante de stabilité pour les éléments finis de Fortin-Soulie

International audienceNous proposons d'analyser la discrétisation du problème de Stokes avec des éléments finis non conformes d'ordre 2 en dimension 2 : les éléments de Fortin-Soulie. Nous évaluons la constante de stabilité dans le cas d'un champ de vitesse peu régulier, ce qui permet d'obtenir des estimations d'erreur a priori dans ce cas. Pour cela, on s'inspire de la preuve du Lemme 4 du papier original de Crouzeix-Raviart, en l'adaptant au cas de l'approximation d'un champ de vitesse peu régulier. Puis nous donnons des résultats numériques dans ce cas, et on montre l'importance d'utiliser une méthode de reconstruction de vitesse à divergence nulle

TrioCFD: code & numerical schemes: E. Jamelot and TrioCFD team

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