Stabilization and Operator Preconditioning of Bulk--Surface CutFEM via Harmonic Extension

TL;DR

This paper introduces a stabilized CutFEM for the Laplace--Beltrami equation on curves coupled with a bulk harmonic problem, achieving uniform conditioning and optimal error estimates without explicit stabilization.

Contribution

It develops a novel coupling approach using harmonic extension via the lattice Green's function to improve conditioning and stability of CutFEM on cut cells.

Findings

Condition number is uniformly bounded regardless of cut-cell size.

The method achieves optimal convergence rates in $H^1$ and $L^2$ norms.

Numerical experiments confirm robustness and optimality.

Abstract

We present a cut finite element method (CutFEM) for the Laplace--Beltrami equation on a smooth closed curve $\Gamma\subset\mathbb{R}^2$ coupled to a harmonic bulk problem in $\Omega$ that requires \emph{no explicit stabilization}: no ghost penalty, normal-gradient penalty, or cell agglomeration. The classical ill-conditioning of trace finite element spaces on cut cells arises from basis functions with vanishingly small support on $\Gamma$; our observation is that coupling the surface discretization to a discrete bulk harmonic extension, realized through the lattice Green's function (LGF) on the background Cartesian grid, rigidly constrains the degrees of freedom responsible for this ill-conditioning. The reduced operator, obtained by a congruence transform of the full CutFEM stiffness, inherits symmetry and positive semi-definiteness from the variational form and has a condition number bounded uniformly in the smallest cut-cell ratio. The direct reconstruction has the standard $O(h^{-2})$ mesh conditioning; the single-layer density formulation acts as operator preconditioner and yields $O(1)$ conditioning, which is amenable to iterative solvers; the double-layer density formulation remains cut-independent with $O(h^{-2})$ scaling. We prove optimal $O(h)$/$O(h^2)$ error estimates in $H^1(\Gamma)$/$L^2(\Gamma)$ under standard regularity assumptions, establish the cut-independent conditioning rigorously, and demonstrate both the optimal convergence rate and robustness with respect to small cuts in numerical experiments.