Stability of Finite-Element Methods for Incompressible Elasticity

Definitions and mathematical statement

The supplied record concerns two benchmark problems in two-dimensional incompressible finite elasticity. It distinguishes continuum solutions from three types of finite-element discretization: displacement, mixed, and mixed-enhanced formulations. Neither benchmark nor any of these discretizations is specified mathematically in the supplied extraction.

The recorded question is:

For each of the two benchmark problems, determine the stability properties of the continuum solutions and of their displacement, mixed, and mixed-enhanced finite-element discretizations.

This is an incomplete problem description, not a fully specified mathematical assertion. The reference domains, constitutive laws, loads, boundary conditions, loading parameters, solution classes, discrete spaces, meshes, enforcement of incompressibility, and meaning of the mixed enhancement are not supplied. The notion of stability, admissible perturbations, and any parameter ranges or uniformity requirements are also unspecified. Consequently, no precise stability criterion or quantified mathematical target can be recovered from this record alone.

Short English statement

The question asks about stability of continuum solutions and three finite-element formulations for two incompressible elasticity benchmarks in two dimensions. The benchmarks, discretizations, and meaning of stability are missing, so the supplied extraction does not determine a precise mathematical problem.

Sources

• Local UnsolvedMath registry, supplied source record: permanent ID $30001991$, source problem code catalogue entry, title Stability of Finite-Element Methods for Incompressible Elasticity; collection: Oberwolfach Reports Open Problems. No individual proposer is identified; the supplied proposed year is 2012.
• Advanced Computational Engineering (2012), Oberwolfach Reports, pp. 457--553, DOI 10.4171/owr/2012/09. Supplied source URL and literature-source link, labelled "original Oberwolfach report": https://doi.org/10.4171/owr/2012/09. The supplied background describes Oberwolfach Reports as documenting mathematical workshops and their open-problem discussions. The supplied assessment identifies this extraction as a benchmark labelled Problem 1; no problem-specific page or contribution author is supplied.
• Literature assessment dated exactly 2026-08-21: the supplied status is open, and the assessment states that this is an incomplete extraction supplying neither a complete mathematical statement nor a resolution. This is supplied dated curation, not independently verified. The supplied background requests independent verification of time-sensitive status and cited claims.
• The registry also labels the statement corrected_verified and records a manual semantic, typographic, and LaTeX review claiming that the cleanup is self-contained and adds no unsupported source-dependent content. That supplied review label does not resolve the missing benchmark specifications or constitute mathematical verification of stability claims.