$\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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| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
| Published: |
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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| Summary: | 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. |
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| ISSN: | 1778-3569 |