Characterize admissible distortion for bounded effectivity

Characterize a necessary and sufficient condition on the damaged region that ensures the effectivity index of equilibrated flux estimators remains bounded, and derive a corresponding bound on the local efficiency constant in terms of the number of flattened cells sharing a vertex patch.

Background

The paper studies equilibrated flux estimators for conforming Lagrange finite element approximations of the Poisson problem on coarse and distorted meshes. Reliability is guaranteed independently of mesh shape, but local efficiency can deteriorate through a shape-regularity constant associated with vertex patches.

Numerical experiments rule out the diameter of the damaged region as the relevant parameter controlling efficiency. Instead, the authors identify the number of flattened cells sharing a single vertex patch as a plausible controlling quantity. The paper explicitly states that converting this observation into a rigorous bound for the local efficiency constant remains unresolved.

References

Section~\ref{sec:numerics_pw} rules out the diameter of the damaged region as the relevant parameter and points instead to the number of flattened cells sharing a vertex patch; turning that observation into a bound on $C_{\mathrm{st}$ is open.

— A posteriori error estimation for coarse or distorted meshes  (2610.12088 - Chouly et al., 8 Oct 2026) in Section 6, subsection “Open directions,” item 1; see also Section 5.4, subsection “The damage-extent conjecture”