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A penalized φ-FEM scheme for the Poisson Dirichlet problem

Raphaël Bulle, Michel Duprez, Vanessa Lleras, Killian Vuillemot

TL;DR

This work analyzes a penalized $\varphi$-FEM for the Poisson equation with Dirichlet data on domains described by a level-set $\varphi$. Boundary conditions are enforced via a penalty term and stabilized with ghost penalties, with the level-set information needed only on boundary-adjacent cells. The authors prove an $H^1$-optimal, $L^2$-quasi-optimal a priori error theory and demonstrate favorable conditioning, validated by 2D and 3D numerical tests that compare against standard FEM and the original $\varphi$-FEM. The results show robust, accurate unfitted discretization for complex geometries and offer practical guidance for implementation and performance in both 2D and 3D settings.

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.

A penalized φ-FEM scheme for the Poisson Dirichlet problem

TL;DR

This work analyzes a penalized -FEM for the Poisson equation with Dirichlet data on domains described by a level-set . Boundary conditions are enforced via a penalty term and stabilized with ghost penalties, with the level-set information needed only on boundary-adjacent cells. The authors prove an -optimal, -quasi-optimal a priori error theory and demonstrate favorable conditioning, validated by 2D and 3D numerical tests that compare against standard FEM and the original -FEM. The results show robust, accurate unfitted discretization for complex geometries and offer practical guidance for implementation and performance in both 2D and 3D settings.

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.
Paper Structure (7 sections, 6 theorems, 39 equations, 5 figures, 1 table)

This paper contains 7 sections, 6 theorems, 39 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proposed scheme and theoretical results
  3. Coercivity of the bilinear form
  4. Proof of the $H^1$ estimate
  5. Numerical illustration
  6. First test case: a 2D complex geometry
  7. Second test case: a 3D geometry

Key Result

Theorem 1

We suppose that assumptions assumption1 and assumption2 on $\Gamma$ and $\mathcal{T}_h$ are satisfied, $f \in H^{k-1}(\Omega_h)$, $\Omega \subset \Omega_h$ and $\gamma,~\sigma_D$ large enough. Consider $u \in {H^{k+1}(\Omega)}$ the solution of eq:main_poisson_equation_dirichlet_homo and $u_h \in V_h with $C>0$ a constant independent on $h$.

Figures (5)

  • Figure 1: Test case 1. From left to right: reference solution (standard FEM with a fine mesh); difference between the reference solution and the projection of each approximated solution (Standard FEM, Direct $\varphi$-FEM, Dual $\varphi$-FEM).
  • Figure 2: Test case 1.$L^2$ (left) and semi-$H^1$ (right) relative errors with respect to the mesh size.
  • Figure 3: Test case 1. Left: computation time with respect to the mesh size. Right: condition number with respect to the mesh size.
  • Figure 4: Test case 2. From left to right: reference solution (standard FEM with a fine mesh); difference between the reference solution and the projection of each approximated solution (Standard FEM, Direct $\varphi$-FEM, Dual $\varphi$-FEM).
  • Figure 5: Test case 2.$L^2$ (left) and semi-$H^1$ (right) relative errors with respect to the mesh size.

Theorems & Definitions (10)

  • Remark 1
  • Theorem 1
  • Proposition 1
  • Lemma 1
  • proof
  • Lemma 2: cf. phifem
  • proof : Proof of Proposition \ref{['lemma:coercivity_dual_phi_fem']}
  • Lemma 3: cf. phiFEM2
  • Lemma 4
  • proof : Proof of Theorem \ref{['thm:convergence_dual_phi_fem']}, $H^1$ estimate