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State and adjoint Strang bounds give a reduced-gradient error certificate

Statement

Assume the continuous problem of Linear coercive elliptic PDE control problem and reduced objective, Countable Choice, and a finite-dimensional conforming subspace $V_{h} \subset V$ . Let $u = u (q)$ and $p = p (q)$ be the exact state and adjoint. Let $\tilde{u}, \tilde{p} \in V_{h}$ be computed state and adjoint, where $\tilde{p}$ uses $\tilde{u}$ in its right-hand side. Suppose the primal and transposed discrete variational problems meet the form-linearity, load-linearity, boundedness and coercivity hypotheses of First Strang estimate for conforming perturbed Galerkin problems. Let $ε_{u}$ be the first-Strang right-hand side for the primal problem, and let $ε_{p}$ be that right-hand side for the adjoint problem when its load defect includes the additional term $∥ J ∥^{2} ε_{u}$ . Then

∥ u - \tilde{u} ∥_{V} \leq ε_{u}, ∥ p - \tilde{p} ∥_{V} \leq ε_{p}, ∥ \nabla \hat{J} (q) - (B^{*} \tilde{p} + α q) ∥_{U} \leq ∥ B ∥ ε_{p} .

The Strang terms retain best-approximation, quadrature/geometry form defects, and load defects; no defect is silently set to zero.

Facts & Assumptions

Given: The exact state and adjoint; the perturbed conforming state and adjoint equations; uniform discrete coercivity; and the primal/adjoint Strang bounds defined in the Statement.

[F1]

The exact state and adjoint satisfy the equations in Linear coercive elliptic PDE control problem and reduced objective and Well-posedness of elliptic control state and adjoint equations.

[F2]

The first Strang estimate is $∥ w - w_{h} ∥_{V} \leq \inf_{v_{h}} ((1 + M / β_{h}) ∥ w - v_{h} ∥_{V} + \frac{δ_{a, h} (v_{h}) + δ_{F, h}}{β_{h}})$ for each bounded coercive real bilinear exact form and its bounded coercive conforming real bilinear discrete form, with bounded linear loads (First Strang estimate for conforming perturbed Galerkin problems).

[F3]

The reduced gradient is $B^{*} p + α q$ and $∥ B^{*} ∥ \leq ∥ B ∥$ (Reduced gradient and coercive Hessian for linear elliptic tracking control).

Proof

1.1F1F2givenalgebra

Apply [F2] to the exact state equation and the computed state equation. By the definition of $ε_{u}$ , this gives $∥ u - \tilde{u} ∥_{V} \leq ε_{u}$ . The adjoint load formed with $\tilde{u}$ differs from the exact adjoint load by the functional $v ⟼ (J (\tilde{u} - u), J v)_{H},$ whose norm is at most $∥ J ∥^{2} ∥ \tilde{u} - u ∥_{V} \leq ∥ J ∥^{2} ε_{u}$ . Including this term in the adjoint load defect and applying [F2] to the transposed form gives $∥ p - \tilde{p} ∥_{V} \leq ε_{p}$ .

2.1F3step 1.1algebra∎

The exact reduced gradient is $B^{*} p + α q$ by [F3], and the computed one is $B^{*} \tilde{p} + α q$ . Therefore $∥ \nabla \hat{J} (q) - (B^{*} \tilde{p} + α q) ∥_{U} = ∥ B^{*} (p - \tilde{p}) ∥_{U} \leq ∥ B ∥ ∥ p - \tilde{p} ∥_{V} \leq ∥ B ∥ ε_{p} .$ The displayed certificate includes every Strang approximation and consistency term used to form $ε_{p}$ .

Remarks

For exact conforming Galerkin equations, the form and load defects vanish and the Strang bounds reduce to the matching Céa estimates. For quadrature or geometry approximations, the corresponding defects remain in $ε_{u}$ and $ε_{p}$ .

Depends on

Used by

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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)