Output error estimation

Let $h$ denote the working space (degree $p$) and $H$ an enriched space (degree $p+1$ on the same mesh). Inject the converged coarse solution, $\mathbf U_H^h = \mathbf I\,\mathbf U_h$. It does not satisfy the fine equations; the fine residual $\mathbf R_{w,H}(\mathbf U_H^h)$ measures by how much. Linearising the fine output about the injected state gives

with $\boldsymbol\psi_H$ the fine-space discrete adjoint linearised about $\mathbf U_H^h$. Gyre solves that one linear system with the p-multigrid hierarchy $(p+1, p, \dots, 0)$ and returns

• value $= J_h$, estimate $=\delta J$, corrected $= J_H(\mathbf U_H^h) + \delta J$, and error $= J_h - $ corrected;
• indicators: $\eta_e = \big|\sum_{i\in e}\boldsymbol\psi_{H,i}\cdot\mathbf R_{w,H,i}\big|$, the share of each element — the quantity an adaptation strategy would act on.

No fine nonlinear solve is needed.

The estimate is the first-order term of a Taylor expansion, so its error is quadratic in the distance between $\mathbf U_H^h$ and $\mathbf U_H$. Its quality hinges on $\boldsymbol\psi_H$ being a good approximation of the continuous adjoint, i.e. on adjoint consistency of the discretization and of the output definition (see BR2 and boundary conditions). For an adjoint-consistent scheme, outputs converge at up to twice the rate of the solution in the energy norm, and the corrected output converges faster still. Singularities (a sharp trailing edge) reduce these rates, as they reduce everything else.

Effectivity is assessed against the actually converged $p+1$ solution — an exact test of the estimator, independent of any “true” solution:

gyre run naca0012_euler --error cd

or request error_estimate: ["cd"] in a job, or call session_error_estimate from an agent. The command prints value, estimate, corrected and error for each output. To keep the per-element indicators, add -o <dir> (the full record is then in result.json) or use a job or a session (the agent tool session_error_estimate returns them summarised).

In a design loop the estimate answers three practical questions: whether the current mesh/order is adequate for the objective at the current design, whether a change in the objective between two designs is larger than its numerical uncertainty, and where resolution is missing.

The estimate exists for the built-in scalar outputs, which can be rebuilt in the enriched space with the same reference quantities; a user-defined functional or f_surface raises KeyError. It is computed at a converged state only: a job, run_case() or gyre run whose solve did not converge returns no estimate. The job result and the result of run_case() list it under skipped; for gyre run that entry is in result.json (with -o <dir>), not in the printed summary.

Order selection turns the estimate into a stopping rule. Given an output and an absolute tolerance, the case is solved at its own degree $p_0$, the error of the output is estimated, and the degree is raised by one until

that is, until |error| of the estimate above is within the tolerance, or the largest allowed degree is reached. Each new degree starts from the interpolated solution of the one below. Usage is in Gradients and error estimates; the order-selection study checks the rule.

• The lowest sufficient degree among those tried. Degrees are tried in ascending order from the case’s own, and the first whose estimate meets the tolerance is returned. Degrees below the starting one are never tried.
• Relative to the next degree. The estimate at degree $p$ predicts the output at $p + 1$ on the same mesh. The tolerance therefore bounds the estimated distance to the $p + 1$ output, not to the exact one. Where enrichment in $p$ captures little of the true error (a singularity, an under-resolved shock, a geometry that the mesh itself misrepresents) the estimate, and with it the selection, is optimistic.
• An estimate, not a bound. $\delta J$ is a linearisation; on a mesh too coarse for it to hold, met: true says that the estimate is within the tolerance, nothing more.
• Only a certified estimate counts. The tolerance is met only by a finite estimate whose adjoint solve converged. If a degree does not converge, after it has also been tried with the full continuation and from the free stream, the selection stops with met: false and the last converged degree.
• Uniform enrichment. Every element gets the same degree. The indicators $\eta_e$ are reported with the estimate but not used; the mesh is not changed.
• One output. The degree that suffices for one output need not suffice for another; other outputs of the result carry no accuracy statement unless their own estimate is requested.
• A discrete choice. The selected degree can change between two nearby designs, and the output then changes by the difference between two discretizations. Inside an optimization, select once and keep the degree fixed.