Low-Regularity Right-Hand-Side Approximation in Adaptive Methods

Definitions and mathematical statement

Retain $\mathcal A^s$ for the solution-approximation class in the source. Let $u$ solve the intended differential or variational problem with corresponding right-hand side $f$, and let $V'$ be the dual space whose norm is intended. The conjectural implication is posed for every exponent $0<s\leq1/2$ and every such solution $u\in\mathcal A^s$.

For an admissible partition $\mathcal T$ of the intended domain $\Omega$, denote by $\mathcal P_0(\mathcal T)$ the functions constant on each partition element. Interpreting "an $n$-element partition" literally, the requested property is \[ \exists C>0\ \forall n\in{\mathbb{Z}}_{>0}\ \exists\mathcal T_n\ \exists f_n\in\mathcal P_0(\mathcal T_n),\qquad |\mathcal T_n|=n,\quad\|f-f_n\|_{V'}\leq Cn^{-s}. \] The constant may depend on the fixed problem data, $u$ and $s$, but must be independent of $n$. The partition and approximant may depend on $n$; no nestedness or use of the same partition as the best approximation to $u$ is asserted. Membership of the piecewise-constant functions in $V'$ must be part of the missing problem setup. If $V$ is a normed test space, its dual norm is \[ \|g\|_{V'}=\sup_{0\ne v\in V}\frac{|\langle g,v\rangle|}{\|v\|_V}. \] The record does not identify $V$, so a particular Sobolev dual norm or boundary condition is not selected.

A best-approximation interpretation of $\mathcal A^s$ ordinarily concerns boundedness of a quantity such as $\sup_n n^s\inf_{v\in\mathcal U_n}\|u-v\|_V$, but neither the discrete classes $\mathcal U_n$, the solution norm nor the size convention is given. This is explanatory notation, not a definition silently assigned to the source's $\mathcal A^s$. Similarly, adaptive-mesh conventions may count at most $n$ elements or $n$ elements added to an initial mesh, and admissible refinements may not have every exact cardinality. The source needs checking before replacing its literal exact-$n$ wording by one of these alternatives.

The assertion asks for existence of data approximants with the same decay exponent, not an algorithm producing them or a uniform bound in terms of an explicitly specified class seminorm. The literature triage checked 2026-08-21 reports this implication conjectural in the low-regularity range and says later AFEM work uses data-approximation assumptions without establishing the full implication. Its open label and source-verification report remain dated metadata, not independently verified conclusions.

Short English statement

Does a solution's approximation rate $n^{-s}$, for $0<s\leq1/2$, imply the same rate for piecewise-constant approximation of its forcing term in the intended dual norm? The partitions, operator, function spaces and approximation-class definition are missing. The quantifiers require one bound independent of $n$, with the approximant allowed to vary with $n$.

Sources

• Local UnsolvedMath registry, permanent ID 30000295, source code catalogue entry, "Low-Regularity Right-Hand-Side Approximation in Adaptive Methods," including original and cleaned statements and verification metadata.
• Mini-Workshop: Convergence of Adaptive Algorithms (2006), Oberwolfach Reports, pp. 2091--2138, DOI 10.4171/owr/2005/37: https://doi.org/10.4171/owr/2005/37. The citation uses 2006, while the DOI and proposed-year metadata use 2005.
• Supplied reference "Stevenson AFEM paper": https://doi.org/10.1007/s10208-005-0183-0.
• Literature triage checked 2026-08-21; registry record updated 2026-09-16. No external source has been visited or independently checked here.