Constraint Functions in $Q_2$–$P_1$ Incompressible Elasticity

Definitions and mathematical statement

The supplied record concerns the incompressible finite-elasticity problem referred to as "Problem 2" in the cited report. Write $\Theta$ for its constraint function and $Q_2\text{--}P_1$ for the finite-element discretization named in the record. The governing problem, the formula and admissible class for $\Theta$, and the precise finite-element spaces are not supplied; these symbols therefore retain their source-specific meanings without additional assumptions.

Research question. For that incompressible finite-elasticity problem and its $Q_2\text{--}P_1$ discretization, determine how the choice of $\Theta$ affects stability and the numerical solution under mesh refinement.

This is an incompletely specified research question. A precise mathematical formulation requires the equations or variational formulation of Problem 2, the spatial domain and dimension, constitutive assumptions, loading and boundary conditions, the role and admissible choices of $\Theta$, the discrete spaces and mesh family, and the intended stability and solution-comparison criteria. In particular, the supplied record does not specify a stability inequality, a convergence topology, or the quantifiers over constraint functions and refining meshes. These qualifications cannot be resolved from the supplied evidence.

Short English statement

The question asks how the constraint function $\Theta$ influences stability and computed solutions for a $Q_2\text{--}P_1$ discretization of incompressible finite elasticity as the mesh is refined. The referenced Problem 2 and the necessary analytical and numerical specifications are missing, so a fully precise statement remains subject to source clarification.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001992$, source problem code catalogue entry, in the Oberwolfach Reports Open Problems collection. The record gives the proposed year as 2012 and names no proposer.
• Advanced Computational Engineering (2012), Oberwolfach Reports, pp. 457--553, DOI 10.4171/owr/2012/09. Supplied source reference and literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/09. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions. No problem-specific page or author is supplied.
• Literature assessment dated 2026-08-21: the supplied curation labels the problem open and reports numerical dependence on the penalty function without a general theorem establishing the claimed behavior. The background identifies the original report as the source checked and describes the assessment as dated literature triage requiring independent verification. This is supplied dated curation, not an independently verified literature assessment; the report has not been consulted in preparing this draft.
• The registry labels the statement corrected_verified and records a manual semantic, typographic, and LaTeX review asserting a self-contained, meaning-preserving cleanup. That supplied editorial assessment does not resolve the missing mathematical specifications identified here.