Optimal Mesh Selection with A Posteriori Error Estimates

Definitions and mathematical statement

Research problem. Develop adaptive error control and optimal mesh-selection theories that use a posteriori estimates allowing cancellation of elementwise errors while preserving relevant stability properties. Here an adaptive method repeatedly computes a discrete PDE solution on a mesh, estimates the error from computed data, and changes the mesh in response. Cancellation means that the error assessment can retain the interaction of signed local contributions, rather than necessarily replacing their global sum by a sum of absolute values. The PDE class, error quantity, approximation spaces, admissible meshes, stability conditions, and meaning of optimality are not specified in the record, so this is a research program rather than a single quantified theorem.

A conditional formulation illustrating the required choices is as follows. After fixing an exact solution $u$, a quantity of interest $J$, an admissible stable mesh class $\mathcal T_{\mathrm{adm}}$, and discrete solutions $u_{\mathcal T}$, seek computable signed contributions $\eta_K$, $K\in\mathcal T$, with rigorous control of the global error $J(u)-J(u_{\mathcal T})$, including any remainder. For example, if a valid representation is $J(u)-J(u_{\mathcal T})=\sum_{K\in\mathcal T}\eta_K+R_{\mathcal T}$, a reliable bound must also control $R_{\mathcal T}$; cancellation in $\sum_K\eta_K$ alone is insufficient. Seek a mesh-selection procedure that achieves a prescribed error tolerance while retaining the fixed stability conditions, together with an optimality or complexity theorem relative to the best meshes in that same admissible stable class. This example specifies neither an error identity for arbitrary PDEs nor a particular optimality notion absent from the source.

The supplied literature triage, dated 2026-08-21, classifies the problem as partially solved: dual-weighted residual and goal-oriented adaptive methods use signed global contributions and can exploit cancellation for selected quantities of interest, while general optimal mesh selection with cancellation and simultaneous stability constraints remains problem-dependent. This dated assessment requires independent verification.

Short English statement

Develop adaptive PDE methods whose error estimates can exploit cancellation between local contributions, with rigorous error control and mesh optimality that preserve the relevant stability conditions. The source leaves the PDE, error norm or output, and stability and optimality criteria open.

Sources

• Local UnsolvedMath registry, permanent ID 30000003, code catalogue entry; title Optimal Mesh Selection with A Posteriori Error Estimates; Oberwolfach Reports Open Problems collection.
• Self-Adaptive Methods for PDE (2005), Oberwolfach Reports, DOI 10.4171/owr/2004/16, pp. 829--872. https://doi.org/10.4171/owr/2004/16. The registry gives proposed year 2004 and tag owr-year:2004; no proposer is named.
• Supplied literature source labeled goal-oriented AFEM with optimal complexity: https://arxiv.org/abs/2101.11407.
• Supplied literature source labeled goal-oriented mesh adaptation: https://www.sciencedirect.com/science/article/abs/pii/S0045782506001137.
• Supplied literature assessment checked 2026-08-21, status partially_solved; record updated 2026-09-16. This is dated triage from the supplied OWR curation, not independently verified here.