$\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space

Michel Duprez; Vanessa Lleras; Alexei Lozinski; Killian Vuillemot

doi:10.5802/crmath.497

Analyse numérique

ϕ

-FEM for the heat equation: optimal convergence on unfitted meshes in space

Michel Duprez ¹ ; Vanessa Lleras ² ; Alexei Lozinski ³ ; Killian Vuillemot ^1,²

¹ MIMESIS team, Inria Nancy - Grand Est, MLMS team, Université de Strasbourg, 1 place de l’hôpital, 67000 Strasbourg, France
² IMAG, Univ Montpellier, CNRS UMR 5149, 499-554 Rue du Truel, 34090 Montpellier, France
³ Université de Franche-Comté, Laboratoire de mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon Cedex, France

Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1699-1710.

Résumé

Thanks to a finite element method, we solve numerically parabolic partial differential equations on complex domains by avoiding the mesh generation, using a regular background mesh, not fitting the domain and its real boundary exactly. Our technique follows the $ϕ$ -FEM paradigm, which supposes that the domain is given by a level-set function. In this paper, we prove a priori error estimates in $l^{2} (H^{1})$ and $l^{\infty} (L^{2})$ norms for an implicit Euler discretization in time. We give numerical illustrations to highlight the performances of $ϕ$ -FEM, which combines optimal convergence accuracy, easy implementation process and fastness.

Reçu le : 2023-02-08
Accepté le : 2023-03-14
Accepté après révision le : 2023-03-21
Publié le : 2023-12-21

DOI : 10.5802/crmath.497

Affiliations des auteurs :

Michel Duprez ¹ ; Vanessa Lleras ² ; Alexei Lozinski ³ ; Killian Vuillemot ^{1, 2}

¹ MIMESIS team, Inria Nancy - Grand Est, MLMS team, Université de Strasbourg, 1 place de l’hôpital, 67000 Strasbourg, France
² IMAG, Univ Montpellier, CNRS UMR 5149, 499-554 Rue du Truel, 34090 Montpellier, France
³ Université de Franche-Comté, Laboratoire de mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon Cedex, France

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Michel Duprez and Vanessa Lleras and Alexei Lozinski and Killian Vuillemot},
     title = {$\phi ${-FEM} for the heat equation: optimal convergence on unfitted meshes in space},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1699--1710},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {361},
     year = {2023},
     doi = {10.5802/crmath.497},
     language = {en},
}

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Michel Duprez; Vanessa Lleras; Alexei Lozinski; Killian Vuillemot. $\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space. Comptes Rendus. Mathématique, Volume 361 (2023), pp. 1699-1710. doi : 10.5802/crmath.497. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.497/

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