Sharpness of the two-cell ghost-weight threshold

Prove whether the cut-uniform ghost-weight threshold for the aligned two-cell CutFEM model equals the vanishing-cut threshold $1/(\gamma_D-p^2)$, thereby establishing the matching upper bound for every cut fraction.

Background

The paper derives the exact semidefiniteness threshold of the vanishing-cut limit of an aligned two-cell model and shows that this threshold is a lower bound for the supremum of the thresholds over all cut fractions. Numerical generalized-eigenvalue computations support equality at sampled cut fractions, but the analysis does not prove that no intermediate cut fraction requires a larger ghost-penalty weight. The matching upper bound is therefore explicitly unresolved.

References

Whether the cut-uniform threshold equals the limit, that is, whether no cut fraction demands a weight beyond $1/(\gamma_D-p2)$, remains unproved: the exact generalized-eigenvalue computations reported with Table~\ref{tab:spd-floor} support it at the sampled cut fractions, and we leave an analytic proof of the matching upper bound open.

— A Multigrid Method for CutFEM and its Convergence  (2609.04067 - Wichrowski, 3 Sep 2026) in Appendix B, immediately following Proposition B.1, “vanishing-cut threshold of the two-cell model”

Because additional ghost faces add stabilization, the one-face model does not certify indefiniteness of the assembled form or explain this GMRES behaviour; the usable weight range and strip contraction rate remain open.

— A Multigrid Method for CutFEM and its Convergence  (2609.04067 - Wichrowski, 3 Sep 2026) in Section 5.2, subsection “Limits of ghost-penalty shedding”