Prove λ-uniform L2-error bounds with anisotropic stabilization on cut subcells

Prove that the anisotropic stabilization weight $h_T^{-1}(2\mu_i+d\lambda_i n_T\otimes n_T)$ yields an $L^2$-error estimate uniform in the Lamé parameter $\lambda$ for the unfitted hybrid high-order method on cut subcells, thereby overcoming the absence of an $H(\operatorname{div})$-conforming interpolant on those subcells.

Background

The paper studies an unfitted hybrid high-order discretization of elastodynamics with a linear-slip imperfect interface. Its standard isotropic stabilization produces errors that grow proportionally to λ1/2\lambda^{1/2} in the quasi-incompressible limit, although the observed convergence rates remain optimal. The authors report that an anisotropic stabilization weight borrowed from the fitted method restores numerical uniformity of the L2L^2-error as λ→∞\lambda\to\infty.

A rigorous proof of this uniformity is unresolved because the fitted analysis uses an H(div⁡)H(\operatorname{div})-conforming, BDM-type interpolant whose normal-trace error vanishes on element faces, whereas the boundaries of unfitted cut subcells include the interface portion TΓT^\Gamma. The paper therefore leaves open the construction or use of a suitable interpolant and the resulting proof of a λ\lambda-uniform L2L^2-error estimate.

References

A proof of the $\lambda$-uniformity of the $L2$-error with the anisotropic weight would require an $H(\dive)$-conforming interpolate on the cut subcells, which is not available; we leave this question to future work and we use the simpler stabilisation eq:stab in all the other experiments.

eq:stab:

$\begin{aligned} s_T^{\partial T}(_T,_T)&:=\sum_{i=1,2}\frac{2\mu_i+d\lambda_i}{h_T} \big\langle{\bf S}(_{T^i}),{\bf S}(_{T^i})\big\rangle_{(\partial T)^i},\\ s_T^\Gamma(v_T,w_T)&:=\big\langleS_h[{v_T}_\Gamma,[{w_T}_\Gamma \big\rangle_{T^\Gamma}, \end{aligned} $

— An Unfitted Hybrid High-Order Method for the Elastodynamics Problem with Imperfect Interface  (2610.01504 - Huang et al., 1 Oct 2026) in Section 7, subsection “The quasi-incompressible limit” (Section 7.4)