Ghost stabilisation for cut finite element exterior calculus

TL;DR

The paper develops CutFEEC, a robust unfitted discretisation framework that merges Finite Element Exterior Calculus with CutFEM to solve Hodge Laplace problems on domains cut by a background mesh. A novel ghost penalty stabilisation yields a global, stable $L^2$-inner product on the active mesh, proving uniform equivalence with the physical domain norm and enabling preservation of the de Rham complex for arbitrary form degree, dimension, and topology. The approach includes a ghost-adapted Hodge decomposition and a stabilized discrete harmonic space, with numerical validation on a filled torus demonstrating convergence and conditioning independent of boundary position. The results provide a practical, scalable method for unfitted FEEC discretisations with provable stability and predictable performance in complex geometries. Overall, CutFEEC offers a principled path to robust, high-order, unfitted simulations of electromagnetic, fluid, and elasticity problems governed by de Rham complexes.

Abstract

We introduce the cut finite element method in the language of finite element exterior calculus, by formulating a stabilisation -- for any form degree -- that makes the method robust with respect to the position of the interface relative to the mesh. We prove that the $L^2$-norm on the physical domain augmented with this stabilisation is uniformly equivalent to the $L^2$-norm on the ``active'' mesh that contains all the degrees of freedom of the finite element space (including those external to the physical domain). We show how this CutFEEC method can be applied to discretize the Hodge Laplace equations on an unfitted mesh, in any dimension and any topology. A numerical illustration is provided involving a conforming finite element space of $H^{\text{curl}}$ posed on a filled torus, with convergence and condition number scaling independent of the position of the boundary with respect to the background mesh.

Figures (4)

• Figure 3.1: Illustration of the active mesh $\mathcal{T}_h$. The boundary $\partial\Omega$ of the domain is in red. The "remainder" of background mesh $\mathcal{T}_{0,h}\setminus \mathcal{T}_h$ is shown in light blue, while the active mesh $\mathcal{T}_h$ is shown in black.
• Figure 5.1: Both $T_1$ and $T_2$ are cut elements, but $T_2$ has a fully immersed neighbour $T_3$. The boundary $\partial\Omega$ is the red curve. Here $N=3$.
• Figure 6.1: Plot of the convergence of the mixed method \ref{['eqs:unfitted_discrete_mixed']} as a function of the mesh size $h$. The convergence is shown for polynomial order $r=1$, and the $1$-norm estimate of the condition number $\kappa_s \coloneq \|\mathcal{A}_s\|_{\mathop{\mathrm{op}}\nolimits} \|\mathcal{A}_s^{-1}\|_{\mathop{\mathrm{op}}\nolimits}$ of the system matrix $\mathcal{A}_s$ is also plotted on the right figure.
• Figure 6.2: Heat map of computed $x$-component of the solution $\eta_h$ to the discrete Hodge Laplace equation \ref{['eqs:unfitted_discrete_mixed']} on the filled in torus. The mesh size is $h=0.019231$ and the polynomial order is $r=1$.

Theorems & Definitions (18)

• Remark 1: Construction of active mesh
• Remark 2: Quasi-uniformity along cut-to-uncut paths
• Lemma 1: Characterisation of the tangential and normal parts
• proof
• Lemma 2
• proof
• Remark 3: Traces and jumps
• Proposition 1: Jump of discrete forms
• proof
• Remark 4: Formula \ref{['eq:ghost_penalty']} for vector proxies