Polynomial dependence of the discrete-extension constant

Determine whether the true constant in the discrete extension estimate for high-order CutFEM functions is polynomial in the polynomial degree p, avoiding the exponential cellwise extrapolation factor.

Background

The discrete extension estimate controls the H1 seminorm on the fictitious portion of the active domain by the physical-domain seminorm and the ghost penalty. The proof propagates polynomial information across chains of cells and introduces an exponential extrapolation factor. The paper states that improving this dependence would require avoiding cellwise extrapolation and explicitly leaves open whether the actual extension constant is polynomial in p.

References

Trace and Markov inequalities and the Cauchy--Schwarz factor $(p+1)\gamma{-1}$ contribute polynomial factors. The chain amplification $\Xi_p$ is exponential in $p$ because each of at most $C_d$ transfer steps incurs the sharp extrapolation constant $C_E(p)\simeq(3+2\sqrt2)p$; improving this factor therefore requires an argument that avoids cellwise extrapolation, and whether the true constant of~eq:ghost-extension is polynomial in $p$ is open.

— A Multigrid Method for CutFEM and its Convergence  (2609.04067 - Wichrowski, 3 Sep 2026) in Remark 4.1, “source of the non-polynomial p-dependence”