Conditional O(h2) variational control error under H2 regularity and full uniform L2 estimates

Assume Countable Choice. Let $q_{\ast }$ minimize the continuous distributed-control objective of Linear coercive elliptic PDE control problem and reduced objective with $E=\mathrm{\Omega }$, $U=H=L^2(\mathrm{\Omega })$, $Bq(v)=(q,v{)}_{L^2}$, and observation $J=I$. Assume exact affine geometry and conforming P1 Galerkin state and adjoint spaces on a shape-regular mesh family. Let $u_h:U\to V_h$ be the affine finite-element state map and define the finite-element reduced objective on $U$ by

$${\widehat{J}}_h(q)=\frac{1}{2}\mathrm{\parallel }{u}_h(q)-u_d{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2+\frac{\alpha }{2}\mathrm{\parallel }q{\mathrm{\parallel }}_{L^2(\mathrm{\Omega })}^2\mathrm{.}$$

Let $p_h(q)\in V_h$ be the associated discrete adjoint. Require adjoint consistency for this same state map and objective, namely

$$D{\widehat{J}}_h(q)[r]=(p_h(q)+\alpha q,r{)}_{L^2(\mathrm{\Omega })}(q,r\in U)\mathrm{.}$$

This identity is the condition that the discrete adjoint represents the derivative of the stated objective; it must be checked for any perturbed state, form, or load implementation. Set $g_h(q)=p_h(q)+\alpha q$ and $g(q)=\mathrm{\nabla }\widehat{J}(q)$, and let $q_h^v$ minimize ${\widehat{J}}_h$ over the same infinite-dimensional box $Q_{ad}$ (variational control discretization). Assume the actual primal and adjoint problems have uniform $H^2$ regularity and the matching primal and dual interpolation bounds of The h^(r+s) L² rate under primal and dual approximation with $r=s=1$. Assume also that the full state and adjoint error bounds, including every Strang form/load defect and the state-error contribution to the adjoint load, satisfy

$$\underset{q\in Q_{ad}}{\sup }(\mathrm{\parallel }{u}_h(q)-u(q){\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }{p}_h(q)-p(q){\mathrm{\parallel }}_{L^2})\le C{h}^2\mathrm{.}$$

The displayed uniform $L^2$ state/adjoint estimate is an explicit hypothesis here; a concrete PDE family must verify it from its actual $H^2$ regularity, matching dual approximation, and complete Strang/geometry/quadrature/load ledger. An energy error of order $h$ alone does not imply this $L^2$ bound. Then

$$\mathrm{\parallel }{q}_h^v-q_{\ast }{\mathrm{\parallel }}_{L^2(E)}\le C{h}^2,$$

where $C$ is independent of $h$. No such rate is asserted on a domain or coefficient class lacking the stated primal, adjoint, and defect estimates.

The $H^2$ primal and adjoint assumptions must match the actual operator, boundary, and domain. In particular, a full-adjoint estimate is not inferred on an arbitrary Lipschitz or reentrant polygonal domain, an energy estimate of order $h$ does not replace the explicit $L^2$ estimate, and unresolved geometry or quadrature defects cannot be discarded.

[F2] Under the matching primal and dual approximation bounds with $r=s=1$, the conforming state error obeys the Aubin–Nitsche $L^2$ estimate with order $h^2$ (The h^(r+s) L² rate under primal and dual approximation, Aubin–Nitsche L² error estimate from adjoint approximation).

[F3] Céa and the shape-regular P1 approximation supply the primal $H^1$ rate used in [F2] (Céa quasi-optimality for coercive conforming Galerkin problems, Primal H¹ approximation on shape-regular affine finite-element meshes).

[F4] The Strang estimate retains every form and load defect for perturbed state and adjoint solves (First Strang estimate for conforming perturbed Galerkin problems).