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Analyse numérique
ϕ-FEM for the heat equation: optimal convergence on unfitted meshes in space
Michel Duprez1   ; Vanessa Lleras2  ; Alexei Lozinski3  ; Killian Vuillemot1, 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.

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 ∞ (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.

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DOI : 10.5802/crmath.497
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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