An elliptic reduced-gradient L2 rate follows under dual approximation

Apply [F1] to the state equation and to the adjoint equation with the transposed coercive form. Under [A1], this yields the two displayed $L^2$ error bounds. Countable Choice is required by [F1].

The parameter-derivative bounds in the Statement give $\mathrm{\mid }{\mathrm{\ell }}_{\theta }^{\mathrm{\prime }}[\eta ](p_h-p)\mathrm{\mid }\le M_{\mathrm{\ell }}^0\mathrm{\parallel }\eta {\mathrm{\parallel }}_2{C}_p{h}^{r+s}$ and, by adding and subtracting $a_{\theta }^{\mathrm{\prime }}[\eta ](u,p_h)$, $\mathrm{\mid }{a}_{\theta }^{\mathrm{\prime }}[\eta ](u_h,p_h)-a_{\theta }^{\mathrm{\prime }}[\eta ](u,p)\mathrm{\mid }\le M_a^0\mathrm{\parallel }\eta {\mathrm{\parallel }}_2(C_u{h}^{r+s}\mathrm{\parallel }{p}_h{\mathrm{\parallel }}_{L^2}+\mathrm{\parallel }u{\mathrm{\parallel }}_{L^2}{C}_p{h}^{r+s})$. The $L^2$ norms of $u$ and $p_h$ are bounded by the stated state bounds and $\mathrm{\parallel }{p}_h{\mathrm{\parallel }}_{L^2}\le \mathrm{\parallel }p{\mathrm{\parallel }}_{L^2}+C_p{h}^{r+s}$.

Apply the exact gradient subtraction in [F2]. The direct-objective and derivative-data defects contribute their assumed $O(h^{r+s})$ bounds; step 2.1 bounds the state- and adjoint-dependent terms by constants times $h^{r+s}$. Taking the supremum over $\mathrm{\parallel }\eta {\mathrm{\parallel }}_2\le 1$ proves the asserted rate. If the $L^2$ derivative bounds or either dual approximation hypothesis fails, this rate has not been proved.