A penalized ϕ-FEM scheme for the Poisson Dirichlet problem

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.

💡 Research Summary

The paper introduces a penalized variant of the φ‑FEM (phi‑FEM) for solving the Poisson equation with homogeneous Dirichlet boundary conditions. Traditional φ‑FEM relies on the representation u = φ w over the whole computational mesh, which automatically satisfies the boundary condition but requires the level‑set function φ (and often its gradient) everywhere, leading to higher preprocessing cost and potential difficulties when φ is not smooth. The authors propose to enforce the boundary condition only on cells intersecting the boundary by introducing an auxiliary variable p defined on the boundary layer Ω_Γ. The solution is expressed as u = φ p, and the relation between u and p is imposed weakly through a penalty term γ/h² ∫_{Ω_Γ}(u − φ p)(v − φ q). In addition, a ghost‑penalty stabilization is added, consisting of face‑wise normal‑derivative jumps and element‑wise Laplacian terms, controlled by a parameter σ_D.

The discrete spaces are standard continuous Lagrange finite‑element spaces V_h^k for u_h and Q_h^k for p_h, restricted to the unfitted mesh T_h that covers the whole computational domain. The bilinear form a_h(u,p;v,q) combines the usual diffusion term, the penalty term, and the ghost‑penalty contributions. Under two geometric assumptions—smoothness of the level‑set function and a patch‑wise covering of the approximate boundary—the authors prove coercivity of a_h with respect to the composite norm

‖(u,p)‖h² = |u|{1,Ω_h}² + h^{−2}‖u − φ p‖{0,Ω_Γ}² + h ∑{F∈F_Γ}‖∂n u‖{0,F}² + h²‖Δu‖_{0,Ω_Γ}².

Key technical tools are two lemmas: one bounding φ_h p_h by the H¹‑seminorm of u_h and the penalty residual, and another providing a trace‑inverse inequality on the boundary layer. With these, Proposition 1 establishes coercivity for sufficiently large γ and σ_D.

The main error result (Theorem 1) shows optimal H¹‑convergence:

|u − u_h|{1,Ω} ≤ C h^k ‖f‖{k−1,Ω_h},

and a quasi‑optimal L²‑estimate of order O(h^{k+½}). The authors note that numerical experiments actually achieve the full O(h^{k+1}) rate in L². They also prove that the condition number of the resulting linear system scales like O(h^{−2}), matching the behavior of other unfitted methods.

Numerical validation is performed on two test cases. The first is a two‑dimensional complex geometry defined by a sum of Gaussian level‑set functions, with source term f = cos(x) exp(y). Three methods are compared: standard fitted FEM, the original “direct” φ‑FEM, and the new “dual” (penalized) φ‑FEM. All three exhibit optimal convergence in both H¹ and L² norms. The penalized φ‑FEM requires the level‑set only on boundary‑adjacent cells, reducing preprocessing effort, and despite the additional unknown p, its total runtime remains comparable to the direct φ‑FEM. Condition numbers remain stable across mesh refinements.

The second test case is a three‑dimensional sphere described by a signed distance level‑set. An analytical solution u_ex = 1 − exp(φ²) is used, guaranteeing zero Dirichlet data on the boundary. With γ = 100 and σ_D = 0.01, the method again shows optimal convergence and well‑behaved conditioning.

Overall, the paper demonstrates that penalizing the boundary condition within the φ‑FEM framework yields a method that is theoretically sound (coercivity, optimal error estimates, predictable conditioning) and practically advantageous (reduced level‑set evaluation domain, comparable computational cost). This contribution broadens the toolbox for unfitted finite‑element methods, offering a flexible and efficient alternative for problems with complex or evolving geometries.