Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2026
|
| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2605.06329 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866913098925867008 |
|---|---|
| author | Xia, Qing |
| author_facet | Xia, Qing |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_06329 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Stabilization and Operator Preconditioning of Bulk--Surface CutFEM via Harmonic Extension Xia, Qing Numerical Analysis Computational Physics 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. |
| title | Stabilization and Operator Preconditioning of Bulk--Surface CutFEM via Harmonic Extension |
| topic | Numerical Analysis Computational Physics |
| url | https://arxiv.org/abs/2605.06329 |