Stabilization and Operator Preconditioning of Bulk--Surface CutFEM via Harmonic Extension
For researchers in computational PDEs, this work provides a stabilization-free CutFEM for bulk-surface problems, addressing the long-standing ill-conditioning issue on cut meshes without heuristic penalties.
The paper presents a cut finite element method for coupled bulk-surface problems that eliminates the need for explicit stabilization (ghost penalty, etc.) by coupling the surface discretization to a discrete bulk harmonic extension via the lattice Green's function. The method achieves optimal convergence rates (O(h) in H^1, O(h^2) in L^2) and uniform conditioning independent of cut-cell size, with the single-layer formulation yielding O(1) condition number.
We present a cut finite element method (CutFEM) for the Laplace--Beltrami equation on a smooth closed curve $Γ\subset\mathbb{R}^2$ coupled to a harmonic bulk problem in $Ω$ 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 $Γ$; 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 $H^1(Γ)$/$L^2(Γ)$ 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.