Uniform Difference-Quotient Estimates on Adaptive Meshes

Definitions and mathematical statement

The source conjectures a local jump estimate uniform over the adaptive mesh family. Given a finite-element mesh $\mathcal T_h$ of $\Omega$ and the Ritz projection $u_h$ of $u\in C^2(\Omega)$, the proposed quantifiers are \[ \forall u\ \exists C(u)<\infty\ \forall h\ \forall K\in\mathcal T_h:\quad \sum_{E\subset\partial K}h_E^{-3} \|[\partial_n u_h]\|_{L^2(E)}^2\leq C(u). \] Here $h_E$ is the diameter/length scale of a mesh face or edge $E$, and the constant is independent of the refinement level and cell, with dependence on fixed problem and mesh-regularity data to be specified. No dependence on uncontrolled mesh grading may be silently hidden in a claimed adaptive-uniform estimate.

For an interior face $E=K^+\cap K^-$, a standard jump convention is \[ [\partial_nu_h]|_E= \nabla u_h|_{K^+}\cdot n_+ +\nabla u_h|_{K^-}\cdot n_-, \] where the normals point out of their respective cells. The sign convention does not affect the squared norm. The source sums over $E\subset\partial K$ without specifying treatment of domain-boundary faces. Ordinary normal derivatives there are not jumps between two cells, and interpreting them by zero exterior extension can change the claimed bound substantially. Neither exclusion nor a boundary convention is silently supplied.

A Ritz projection requires a chosen energy space $V$, finite-element space $V_h\subset V$ and coercive bilinear form $a$, with \[ u_h\in V_h,\qquad a(u_h,v_h)=a(u,v_h)\quad\forall v_h\in V_h. \] The excerpt does not specify $a$, boundary conditions, element degree, spatial dimension, mesh conformity or shape regularity. It also says $C^2(\Omega)$ rather than bounded $C^2(\overline\Omega)$ regularity; membership in the energy space and any global bounds needed for a finite constant must be checked. The displayed projection definition is a framework for the missing data, not an implicit choice of the Laplace or piecewise-linear problem.

The supplied literature assessment checked 21 August 2026 says the report proves the estimate on quasi-uniform meshes but poses the general adaptive estimate as a conjecture, with no general proof or counterexample located. Quasi-uniformity is a special-case qualification, not an added hypothesis resolving the requested adaptive case.

Short English statement

Does the Ritz projection of a smooth function have uniformly bounded cellwise, scaled normal-derivative jumps on adaptive meshes? The constant must work for every mesh level and cell. The source omits mesh, boundary-face and projection conventions; the supplied assessment reports only a quasi-uniform-mesh result.

Sources

• Local UnsolvedMath registry, permanent ID 30000294, exact source code catalogue entry; Uniform Difference-Quotient Estimates on Adaptive Meshes. Proposed year 2005, status open; corrected_verified statement supplies the $h_E^{-3}$ jump inequality and the Ritz projection.
• Supplied report citation: Mini-Workshop: Convergence of Adaptive Algorithms (2006), Oberwolfach Reports, DOI 10.4171/owr/2005/37, pp. 2091--2138, https://doi.org/10.4171/owr/2005/37. Title year differs from DOI/proposed year.
• Supplied literature assessment checked 2026-08-21: "original OWR conjecture", https://doi.org/10.4171/owr/2005/37; "later OWR reproduction", https://ems.press/content/serial-article-files/46016. This dated OWR-derived curation requires independent verification of time-sensitive status and cited claims.