A penalized ϕ-FEM scheme for the Poisson Dirichlet problem
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866908815793848320 |
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| author | Bulle, Raphaël Duprez, Michel Lleras, Vanessa Vuillemot, Killian |
| author_facet | Bulle, Raphaël Duprez, Michel Lleras, Vanessa Vuillemot, Killian |
| contents | In this work, we analyze a penalized variant of the ϕ-FEM scheme for the Poisson equation with Dirichlet boundary conditions. The ϕ-FEM is a recently introduced unfitted finite element method based on a level-set description of the geometry, which avoids the need for boundary-fitted meshes. Unlike the original ϕ-FEM formulation, the method proposed here enforces boundary conditions through a penalization term. This approach has the advantage that the level-set function is required only on the cells adjacent to the boundary in the variational formulation. The scheme is stabilized using a ghost penalty technique. We derive a priori error estimates, showing optimal convergence in the H1 semi-norm and quasi-optimal convergence in the L2 norm under suitable regularity assumptions. Numerical experiments are presented to validate the theoretical results and to compare the proposed method with both the original ϕ-FEM and the standard fitted finite element method. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_05698 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A penalized ϕ-FEM scheme for the Poisson Dirichlet problem Bulle, Raphaël Duprez, Michel Lleras, Vanessa Vuillemot, Killian Numerical Analysis 65N30, 65N85, 65N15 In this work, we analyze a penalized variant of the ϕ-FEM scheme for the Poisson equation with Dirichlet boundary conditions. The ϕ-FEM is a recently introduced unfitted finite element method based on a level-set description of the geometry, which avoids the need for boundary-fitted meshes. Unlike the original ϕ-FEM formulation, the method proposed here enforces boundary conditions through a penalization term. This approach has the advantage that the level-set function is required only on the cells adjacent to the boundary in the variational formulation. The scheme is stabilized using a ghost penalty technique. We derive a priori error estimates, showing optimal convergence in the H1 semi-norm and quasi-optimal convergence in the L2 norm under suitable regularity assumptions. Numerical experiments are presented to validate the theoretical results and to compare the proposed method with both the original ϕ-FEM and the standard fitted finite element method. |
| title | A penalized ϕ-FEM scheme for the Poisson Dirichlet problem |
| topic | Numerical Analysis 65N30, 65N85, 65N15 |
| url | https://arxiv.org/abs/2602.05698 |