Comptes Rendus
no. 7
Numerical analysis
A new fictitious domain method: Optimal convergence without cut elements
[Une nouvelle méthode de type domaine fictif : convergence optimale sans éléments coupés]

Présenté par : Olivier Pironneau

Alexei Lozinski1
1 Laboratoire de mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France
Comptes Rendus. Mathématique, Volume 354 (2016) no. 7, pp. 741-746.

Nous présentons une méthode de type domaine fictif pour le problème de Poisson–Dirichlet. Le maillage de calcul est construit à partir d'un maillage ambiant (typiquement uniforme cartésien) en rejetant les éléments en dehors du domaine dans lequel le problème est posé. Le maillage ainsi obtenu n'est pas ajusté à la frontière du domaine du problème. Plusieurs méthodes d'éléments finis (XFEM, CutFEM) adaptées à ce type de maillages ont été proposées récemment. L'originalité de la méthode que l'on propose ici réside dans le fait que l'on évite l'intégration sur les éléments coupés par la frontière du domaine du problème, tout en préservant le taux de convergence optimal. Cette observation est confirmée par une étude théorique et par des essais numériques.

We present a method of the fictitious domain type for the Poisson–Dirichlet problem. The computational mesh is obtained from a background (typically uniform Cartesian) mesh by retaining only the elements intersecting the domain where the problem is posed. The resulting mesh does not thus fit the boundary of the problem domain. Several finite element methods (XFEM, CutFEM) adapted to such meshes have been recently proposed. The originality of the present article consists in avoiding integration over the elements cut by the boundary of the problem domain, while preserving the optimal convergence rates, as confirmed by both the theoretical estimates and the numerical results.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2016.02.002

Alexei Lozinski 1

1 Laboratoire de mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16, route de Gray, 25030 Besançon cedex, France 
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Alexei Lozinski. A new fictitious domain method: Optimal convergence without cut elements. Comptes Rendus. Mathématique, Volume 354 (2016) no. 7, pp. 741-746. doi : 10.1016/j.crma.2016.02.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.02.002/

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