$\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space
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 $\phi $-FEM paradigm, which supposes t...
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Académie des sciences
2023-12-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.497/ |
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| author | Duprez, Michel Lleras, Vanessa Lozinski, Alexei Vuillemot, Killian |
| author_facet | Duprez, Michel Lleras, Vanessa Lozinski, Alexei Vuillemot, Killian |
| author_sort | Duprez, Michel |
| collection | DOAJ |
| description | 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 $\phi $-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 $\phi $-FEM, which combines optimal convergence accuracy, easy implementation process and fastness. |
| format | Article |
| id | doaj-art-68ef8131cf4644338b5a95915c8cfddb |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-68ef8131cf4644338b5a95915c8cfddb2025-02-07T11:12:14ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-12-01361G111699171010.5802/crmath.49710.5802/crmath.497$\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in spaceDuprez, Michel0https://orcid.org/0000-0002-2059-2811Lleras, Vanessa1https://orcid.org/0000-0003-1358-9558Lozinski, Alexei2https://orcid.org/0000-0003-0745-0365Vuillemot, Killian3MIMESIS team, Inria Nancy - Grand Est, MLMS team, Université de Strasbourg, 1 place de l’hôpital, 67000 Strasbourg, FranceIMAG, Univ Montpellier, CNRS UMR 5149, 499-554 Rue du Truel, 34090 Montpellier, FranceUniversité de Franche-Comté, Laboratoire de mathématiques de Besançon, UMR CNRS 6623, 16 route de Gray, 25030 Besançon Cedex, FranceIMAG, Univ Montpellier, CNRS UMR 5149, 499-554 Rue du Truel, 34090 Montpellier, France; MIMESIS team, Inria Nancy - Grand Est, MLMS team, Université de Strasbourg, 1 place de l’hôpital, 67000 Strasbourg, FranceThanks 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 $\phi $-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 $\phi $-FEM, which combines optimal convergence accuracy, easy implementation process and fastness.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.497/ |
| spellingShingle | Duprez, Michel Lleras, Vanessa Lozinski, Alexei Vuillemot, Killian $\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space Comptes Rendus. Mathématique |
| title | $\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space |
| title_full | $\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space |
| title_fullStr | $\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space |
| title_full_unstemmed | $\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space |
| title_short | $\phi $-FEM for the heat equation: optimal convergence on unfitted meshes in space |
| title_sort | phi fem for the heat equation optimal convergence on unfitted meshes in space |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.497/ |
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