TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Piecewise-constant control with shape-regular finite elements has O(h) error

Statement

Assume Countable Choice and the elliptic control problem of Linear coercive elliptic PDE control problem and reduced objective on a shape-regular affine simplicial mesh family, with cellwise-constant box bounds and P1 conforming state and adjoint spaces. Let $U_{h} \subset U$ be the cellwise-constant control space and $Q_{h} = U_{h} \cap Q_{a d}$ . For each $q \in U$ , let $u_{h} (q) \in V_{h}$ be the chosen affine finite-element state map, and define the finite-element reduced objective on $U$ by

{\hat{J}}_{h} (q) = \frac{1}{2} ∥ J u_{h} (q) - u_{d} ∥_{H}^{2} + \frac{α}{2} ∥ q ∥_{U}^{2} .

Let $p_{h} (q) \in V_{h}$ be its associated discrete adjoint. Require adjoint consistency for this same state map and objective, namely

D {\hat{J}}_{h} (q) [r] = (B^{*} p_{h} (q) + α q, r)_{U} (q, r \in U) .

Write $g_{h} (q) = B^{*} p_{h} (q) + α q$ and $g (q) = \nabla \hat{J} (q)$ . Let $q_{*}$ be the continuous minimizer and $q_{h}$ the minimizer of ${\hat{J}}_{h}$ over $Q_{h}$ . Assume $q_{*} \in H^{1} (E)$ , the projected adjoints satisfy $∥ B^{*} p_{h} (q_{h}) ∥_{H^{1} (E)} \leq C_{p}$ uniformly in $h$ , and the exact state/adjoint finite-element gradient error obeys

\sup_{q \in Q_{h}} ∥ g_{h} (q) - g (q) ∥_{U} \leq C_{g} h .

The last bound is supplied by State and adjoint Strang bounds give a reduced-gradient error certificate when its state and adjoint approximations are the same $u_{h}$ and $p_{h}$ used in ${\hat{J}}_{h}$ and the stated primal/adjoint P1 approximation errors and every Strang form/load defect are $O (h)$ uniformly over $Q_{h}$ . Then there is a constant $C$ independent of $h$ such that

∥ q_{h} - q_{*} ∥_{L^{2} (E)} \leq C h .

Facts & Assumptions

Given: The unique continuous minimizer and its variational inequality; the discrete objective and adjoint-consistency identity in the Statement; its minimizer on $Q_{h}$ ; the cell-average projection $Π_{h}$ ; $q_{*} \in H^{1} (E)$ ; the uniform projected-adjoint $H^{1}$ bound; and the gradient error bound in the Statement.

[F1]

The continuous gradient $g (q) = \nabla \hat{J} (q)$ is strongly monotone: $(g (q) - g (r), q - r)_{U} = ∥ S (q - r) ∥_{H}^{2} + α ∥ q - r ∥_{U}^{2} \geq α ∥ q - r ∥_{U}^{2}$ (Reduced gradient and coercive Hessian for linear elliptic tracking control).

[F2]

The continuous minimizer satisfies $(g (q_{*}), v - q_{*})_{U} \geq 0$ for all $v \in Q_{a d}$ (Existence, uniqueness, and variational inequality for elliptic PDE control). Since $q_{h}$ minimizes the differentiable objective ${\hat{J}}_{h}$ on the convex set $Q_{h}$ , the adjoint-consistency identity gives $(g_{h} (q_{h}), v_{h} - q_{h})_{U} \geq 0$ for every $v_{h} \in Q_{h}$ .

[F3]

$Π_{h} q_{*} \in Q_{h}$ , $∥ q_{*} - Π_{h} q_{*} ∥_{U} \leq C_{Π} h ∥ q_{*} ∥_{H^{1} (E)}$ , and $q_{*} - Π_{h} q_{*}$ is orthogonal to every element of $Q_{h}$ (Cell averaging preserves constant box bounds and has an H1 error bound).

[F4]

The adjoint-consistency identity gives $g_{h} (q) = B^{*} p_{h} (q) + α q$ for the same objective. Its error obeys $∥ g_{h} (q_{h}) - g (q_{h}) ∥_{U} \leq C_{g} h$ , and $∥ B^{*} p_{h} (q_{h}) - Π_{h} (B^{*} p_{h} (q_{h})) ∥_{U} \leq C_{Π} h C_{p}$ (State and adjoint Strang bounds give a reduced-gradient error certificate, Cell averaging preserves constant box bounds and has an H1 error bound).

[A1]

The dual regularity, uniform primal/adjoint approximation, and Strang defects are such that the two bounds in [F4] hold uniformly as $h ↓ 0$ .

Proof

1.1F1F2F3F4givenalgebra

Put $e = q_{h} - q_{*}$ . By [F1], $α ∥ e ∥_{U}^{2} \leq (g (q_{h}) - g (q_{*}), e)_{U} .$ Replace $g (q_{h})$ by $g_{h} (q_{h})$ at a cost at most $C_{g} h ∥ e ∥_{U}$ by [F4]. The continuous variational inequality in [F2] gives $- (g (q_{*}), e)_{U} \leq 0$ . The discrete variational inequality with the feasible test $Π_{h} q_{*}$ gives $(g_{h} (q_{h}), e)_{U} \leq (g_{h} (q_{h}), Π_{h} q_{*} - q_{*})_{U} .$ Since $g_{h} (q_{h}) = B^{*} p_{h} (q_{h}) + α q_{h}$ and $q_{h} \in Q_{h}$ , orthogonality in [F3] cancels the regularization term. Orthogonality also lets us subtract $Π_{h} (B^{*} p_{h} (q_{h}))$ from the remaining factor. Hence $α ∥ e ∥_{U}^{2} \leq C_{Π}^{2} C_{p} ∥ q_{*} ∥_{H^{1}} h^{2} + C_{g} h ∥ e ∥_{U} .$

2.1step 1.1algebra∎

Since every mesh cell is a nondegenerate simplex, $h > 0$ . Let $t = ∥ e ∥_{U} / h$ and divide the inequality in step 1.1 by $h^{2}$ to obtain $α t^{2} - C_{g} t - C_{Π}^{2} C_{p} ∥ q_{*} ∥_{H^{1}} \leq 0.$ Thus $t$ is at most the positive root $\frac{C_{g} + \sqrt{C_{g}^{2} + 4 α C_{Π}^{2} C_{p} ∥ q_{*} ∥_{H^{1}}}}{2 α},$ which is independent of $h$ . The resulting upper bound for $t$ is independent of $h$ , proving the $O (h)$ estimate.

Remarks

The finite-element contribution is carried by the exact gradient residual bound. For exact geometry and Galerkin forms it reduces to the matching primal and adjoint energy estimates; a variational crime contributes its full Strang form and load defects. The $H^{1}$ control regularity is essential to the cell-average term.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

46 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Eduardo Casas and Fredi Tröltzsch, Error estimates for linear-quadratic elliptic control problems (standard reference, not scraped)
Michael Hinze, A Variational Discretization Concept in Control Constrained Optimization (standard reference, not scraped)