Uniformly Bounded Cochain Projections for p-Finite Elements

Definitions and mathematical statement

The supplied record asks about cochain projections for the $p$-version finite-element method and reports the conjecture that such projections do not exist in general. It does not specify the domain, boundary conditions, or admissible finite-element subcomplexes. The following gives a precise template for the requested property; these unspecified data must be fixed before it becomes a fully determined problem.

For a chosen domain $\Omega\subseteq{\mathbb{R}}^n$, let $H^k$ be the relevant $L^2$ space of differential $k$-forms, and let \[ d^k:D^k\subseteq H^k\longrightarrow H^{k+1},\qquad 0\leq k<n, \] be the exterior derivative with the chosen operator domains and boundary conditions. Assume these spaces form a de Rham complex: $d^k(D^k)\subseteq D^{k+1}$ and $d^{k+1}d^k=0$ wherever applicable. Set $D^n=H^n$. For polynomial-degree indices $p\in{\mathbb{N}}_{\geq1}$, a conforming finite-dimensional subcomplex consists of spaces $V_p^k\subseteq D^k$ satisfying \[ d^k(V_p^k)\subseteq V_p^{k+1}\qquad(0\leq k<n). \] Here $p$ indexes increasing polynomial degree in the $p$-version; the precise polynomial spaces and mesh are not supplied.

A cochain projection onto this subcomplex is a collection of linear maps \[ \pi_p^k:D^k\longrightarrow V_p^k\qquad(0\leq k\leq n) \] such that \[ \pi_p^k v=v\quad(v\in V_p^k),\qquad d^k\pi_p^k u=\pi_p^{k+1}d^k u\quad(u\in D^k,\ 0\leq k<n). \] The requested $p$-uniform $L^2$ boundedness means that, for the chosen family of subcomplexes, there exist such maps and a constant $C<\infty$, independent of $p$, with \[ \|\pi_p^k u\|_{L^2(\Omega)}\leq C\|u\|_{L^2(\Omega)} \qquad (p\geq1,\ 0\leq k\leq n,\ u\in D^k). \] For fixed finite $n$, using one constant for all degrees is equivalent to allowing a separate $p$-independent constant in each degree. If the operator domains are dense in their respective $L^2$ spaces, these bounds also give bounded extensions to those spaces.

Question and supplied conjecture. For the intended suitable $p$-version finite-element subcomplexes, do cochain projections satisfying the displayed identities and bound exist? The original statement reports the conjecture that they do not exist in general. Interpreted as failure of a general existence assertion, this would require an admissible family for which no such uniformly bounded collection exists; it does not assert failure for every family. Because the admissible class and analytic setting are absent, this interpretation remains a formulation template requiring review.

Short English statement

Can one project onto the intended finite-element de Rham subcomplexes while commuting with the exterior derivative and keeping the $L^2$ operator norms bounded independently of polynomial degree $p$? The supplied record reports a conjecture of nonexistence in general, but leaves the domain, boundary conditions, and admissible families unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30001420$, source problem code catalogue entry, entitled Uniformly Bounded Cochain Projections for p-Finite Elements, in the Oberwolfach Reports Open Problems collection. The record assigns proposed year 2010 and names no individual proposer.
• Computational Electromagnetism and Acoustics (2010), Oberwolfach Reports, pp. 415--519, DOI 10.4171/owr/2010/10. Supplied report reference: https://doi.org/10.4171/owr/2010/10. The supplied background describes Oberwolfach Reports as records of mathematical workshops and their open-problem discussions.
• Boffi--Costabel--Dauge--Demkowicz--Hiptmair, 2011, as identified in the supplied literature sources: https://costabel.perso.math.cnrs.fr/publis/BoCoDaDeHi_pub.pdf. No full bibliographic title or precise supporting passage is supplied.
• Supplied literature assessment, checked on 2026-08-21: the record labels the problem open, reports substantial progress on $p$-version discrete compactness, and states that the cited primary analysis still describes the existence of $p$-uniformly $L^2$-bounded cochain projections on suitable de Rham subcomplexes as open. This is supplied dated curation, not independently verified literature assessment; the background expressly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and reports comparison with the official Oberwolfach report, with notation, context, and question form normalized without changing the source-level claim. This is supplied verification metadata, not verification performed for this draft.