Mesh Structures Behind Even-Odd Superconvergence

Definitions and mathematical statement

Let $\Gamma\subset{\mathbb{R}}^3$ be a smooth two-dimensional surface. Let $\widehat\Gamma_h$ denote a reference triangulation in a uniformly refined family, indexed by a mesh parameter $h$, and let $\Phi_h^k$ be a piecewise polynomial parametrization of degree $k$. The discrete surface is \[ \Gamma_h^k=\Phi_h^k(\widehat\Gamma_h). \] Here $k$ is the polynomial degree of the geometry parametrization; the supplied record does not specify the degrees of the velocity and pressure approximation spaces. Consider a surface finite-element discretization of the surface Stokes equations on these discrete surfaces.

Research question. Characterize the internal mesh structures necessary for the reported even--odd superconvergence phenomenon: enhanced convergence observed for even geometry degree $k$, but not for odd geometry degree $k$, in the settings considered by the source. Determine whether the source's structure of mirrored pairs of triangles is one representative mechanism or the principal mechanism underlying this phenomenon. Explain also how the observed parity dependence can occur in settings where such a mirrored-pair structure is not explicitly visible.

Unresolved formulation qualifications. The record does not define a mirrored pair, identify the error quantity or norm exhibiting superconvergence, state the baseline and enhanced convergence rates, or specify the finite-element spaces and discrete Stokes formulation. It also does not specify whether $\Gamma$ has a boundary, the boundary conditions if applicable, or the regularity assumptions on the Stokes data and solution. Consequently, this is a research question about a reported phenomenon, not a quantified assertion that every even-degree discretization superconverges or that every odd-degree discretization fails to do so. Necessary or sufficient mesh conditions cannot be stated precisely from this record alone. The original excerpt contains damaged adjacent estimates involving undefined curvature-related notation and lift notation; those fragments cannot supply reliable additional hypotheses or conclusions without review of the cited report.

Short English statement

Identify which mesh structures explain the reported difference between even and odd geometry degrees in surface Stokes finite elements. In particular, determine the role of mirrored triangle pairs and explain the same observed effect when no such pairing is visible. The supplied record leaves the relevant error measure, rates, discretization details, and meaning of a mirrored pair unresolved.

Sources

• Local UnsolvedMath registry: supplied complete record for permanent ID $30006576$, Mesh Structures Behind Even--Odd Superconvergence, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives proposed year 2026 and identifies no individual proposer.
• Supplied primary reference and literature-source link: Numerical Analysis for Geometric and Nonlinear PDEs (2026), Oberwolfach Reports, Report 5/2026, pp. 277--344, DOI 10.4171/owr/2026/5: https://doi.org/10.4171/owr/2026/5. The supplied original excerpt includes the report page marker 313.
• Supplied literature assessment, checked 2026-08-22: the record labels the problem open and reports that the workshop report presents both the necessary mesh structure and the explanation without visible mirrored pairs as central open questions; it states that the displayed analysis is not a resolution. This is supplied dated curation, not an independently verified literature assessment.
• The registry labels its statement corrected_verified and reports that the official report was checked and its notation repaired. These are supplied provenance claims; no independent source consultation or mathematical verification is asserted here.