Adaptive Diffusion Parameters for Discretization Accuracy

Definitions and mathematical statement

The source asks: "Can the accuracy of a discretization be enhanced by locally adapting diffusion parameters?" This is a methodological question, not a fully specified universal estimate. It does not identify the diffusion equation, physical domain, boundary conditions, discretization, admissible coefficient changes, or accuracy measure.

A parameterized formulation can expose the missing data without choosing them. Fix a target solution $u$ of the intended original problem, a discrete approximation $u_h(\theta)$ depending on diffusion parameters $\theta\in\Theta_h$, a baseline $\theta_0$, and an error measure $\mathcal E(u,u_h(\theta))$. Local adaptation means changing selected spatial components of $\theta$, with a specified admissible local-update set $\Theta_h^{\mathrm{loc}}(\theta_0)$. The improvement sought would be an admissible choice $\theta_1$ such that \[ \theta_1\in\Theta_h^{\mathrm{loc}}(\theta_0),\qquad \mathcal E(u,u_h(\theta_1))<\mathcal E(u,u_h(\theta_0)). \] For a goal-oriented formulation the error could instead be $|J(u)-J(u_h(\theta))|$ for a specified output $J$; a norm error is another possible choice. Neither is selected by the source. An algorithmic version would require a computable adaptation rule and assumptions guaranteeing such improvement or an appropriate convergence result. The source does not say whether it asks for some useful examples, a class of equations, or an always-improving method.

It is also essential to distinguish numerical diffusion or stabilization parameters from physical model coefficients: if the latter are changed, error must still be measured against the intended original target, with modeling error accounted for, rather than silently redefining $u$ to fit the changed model.

The supplied literature triage, checked 21 August 2026, reports that goal-oriented a posteriori methods support simultaneous model and mesh adaptation, including heterogeneous and nonlinear diffusion models, but does not identify a general theorem guaranteeing that local diffusion-parameter changes always improve a requested output. This is dated supplied curation and does not establish a universal positive or negative answer.

Short English statement

The question asks whether local changes to diffusion parameters can improve a numerical discretization. A precise guarantee needs the original PDE, allowed parameter changes and an error or output criterion. The supplied August 2026 assessment reports model-and-mesh adaptation progress, with no general always-improving guarantee.

Sources

• Local UnsolvedMath registry, permanent ID 30000002, exact source code catalogue entry, Adaptive Diffusion Parameters for Discretization Accuracy; source question retained as supplied. Registry proposed year: 2004; status: partially_solved.
• Supplied report citation: 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 title year and DOI/proposed year differ in the supplied metadata; no silent correction is made.
• Supplied literature assessment checked 2026-08-21: "duality-based model and mesh adaptivity", https://doi.org/10.1515/jnma-2014-0074; "a posteriori control of modeling and discretization errors", https://epubs.siam.org/doi/abs/10.1137/S1540345902410482. This dated OWR-derived triage requests independent verification of time-sensitive status and cited claims.