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arXiv · 2602.05698

A penalized ϕ-FEM scheme for the Poisson Dirichlet problem

Raphaël Bulle·Michel Duprez·Vanessa Lleras·Killian Vuillemot
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Abstract

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.

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Raphaël Bulle, Michel Duprez, Vanessa Lleras, Killian Vuillemot. 2026-02-05. A penalized ϕ-FEM scheme for the Poisson Dirichlet problem. https://arxiv.org/abs/2602.05698

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