--- title: Output error estimation summary: The adjoint that provides gradients also measures how much each element's unresolved residual contaminates an output. One extra linear solve yields an error estimate, a corrected output and per-element indicators. section: theory order: 9 level: 3 keywords: [dual-weighted residual, adjoint-weighted residual, error estimate, superconvergence, adaptation] --- A design optimizer exploits whatever errors the analysis has. Knowing the discretization error of the *objective* — not of the solution in some norm — is therefore the relevant quality measure, and the adjoint provides it. **[L3 Mechanics] The estimate** 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 $$ \delta J \;\equiv\; J_H(\mathbf U_H) - J_H(\mathbf U_H^h) \;\approx\; -\boldsymbol\psi_H^{\mathsf T}\,\mathbf R_{w,H}(\mathbf U_H^h), $$ 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. **[L4 Theory] Why it works, and when** 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](viscous-br2.md) and [boundary conditions](boundary-conditions.md)). 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: **Table: Adjoint-weighted-residual estimate versus the p + 1 solution** (Output error estimation) | mesh | p | cd (p) | estimate δ | corrected | cd (p + 1, converged) | effectivity | remaining / original error | |---|---|---|---|---|---|---|---| | 12x4 | 1 | 0.0177275 | -0.00857008 | 0.00634203 | 0.00569478 | 0.929779 | 0.0537907 | | 24x8 | 1 | 0.012233 | -0.00806656 | 9.2333e-04 | 3.4443e-04 | 0.93304 | 0.0486938 | | 12x4 | 2 | 0.00569478 | -0.0026344 | 5.0612e-04 | 1.4077e-04 | 0.878207 | 0.065781 | | 24x8 | 2 | 3.4443e-04 | -2.0704e-04 | 2.1391e-05 | 1.4254e-05 | 0.966676 | 0.0216164 | Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T11:33:17-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [output_error.json](../data/verification/output_error.json). **Chart: Drag error: raw and adjoint-corrected** (Output error estimation). Convergence chart of |cd − reference| against h (1 / nx), x axis log, y axis log; 4 series. The table holds the plotted data. Reference: finest mesh at degree 4, a solution used by no plotted point and no estimate. It differs from the degree-3 output on the same mesh by 1.4e-05; the reference is itself a computed value, and differences of that size or below are not resolved. Corrected outputs are the injected p + 1 output plus the adjoint estimate. | h (1 / nx) | p = 1: cd | p = 2: cd | p = 1: corrected | p = 2: corrected | |---|---|---|---|---| | 0.0833333 | 0.0177271 | 0.00569443 | 0.00634168 | 5.0577e-04 | | 0.0416667 | 0.0122327 | 3.4408e-04 | 9.2298e-04 | 2.1043e-05 | Provenance: gyre 0.4.0; revision `39979fcf90` (commit `39979fcf902ad6116eb550b881b4278d4599c8e2`); 2026-11-12T11:33:17-0500; device cpu; quick suite. **Revision check: Same code as this build.** Produced by 39979fcf902ad6116eb550b881b4278d4599c8e2. Every commit between it and the commit this site was built from (ceb1f4b0bed41de53e586df10858079f5a2bcdd8) changes only documentation text, evidence records or bookkeeping: the code is the same. Full record: [output_error.json](../data/verification/output_error.json). **[L2 Practice] How to use it** ```bash 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 ` (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 `), not in the printed summary. **[L2 Practice] Choosing the degree from the estimate** 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 $$ |J_h - (J_H(\mathbf U_H^h) + \delta J)| \le \text{tolerance}, $$ 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](../guide/gradients.md#letting-the-error-estimate-choose-the-order); the [order-selection study](../verification/order-selection.md) checks the rule. **[L3 Mechanics] What the selected degree does and does not guarantee** * **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.