will introduce a non-conforming dual-corrected (NCD-corrected) approach; cf. Section \[sec:ncd\_approach\].

FOM for the optimality system {#sec:problem_fom}
-----------------------------

For the discretization of the optimization problem we assume that a finite-dimensional subspace $V_h \subset V$ is given and obtain the FOM for the optimality system of by Ritz-Galerkin projection of equations onto $V_h$. In particular, we have for each $\mu \in {\mathcal{P}}$ the solution $u_{h, \mu} \in V_h$ of the *discrete primal equation* $$\begin{aligned}
{a_{\mu}}(u_{h, \mu}, v_h) = {l_{\mu}}(v_h) &&\text{for all } v_h \in V_h,
\label{eq:state_h}\end{aligned}$$ and hence ${r_{\mu}}^{\textnormal{pr}}(u_{h, \mu})[v_h] = 0$ for all $v_h \in V_h$, $\mu \in {\mathcal{P}}$. We also have for each $\mu \in {\mathcal{P}}$ the solution $p_{h, \mu} \in V_h$ of the *discrete dual equation* $$\begin{aligned}
{a_{\mu}}(v_h, p_{h, \mu}) = \partial_u {\mathcal{J}}(u_{h, \mu}, \mu)[v_h] = {j_{\mu}}(v_h) + 2 {k_{\mu}}(v_h, u_{h, \mu}) &&\text{for all } v_h \in V_h,
\label{eq:dual_solution_h}\end{aligned}$$ and hence ${r_{\mu}}^{\textnormal{du}}(u_{h, \mu}, p_{h, \mu})[v_h] = 0$ for all $v_h \in V_h$, $\mu \in {\mathcal{P}}$. Similarly, the *discrete primal sensitivity equations* for solving for $d_{\nu} u_{h, \mu} \in V_h$ as well as *discrete dual sensitivity equations* for solving for $d_{\nu} p_{h, \mu} \in V_h$ at any direction $\nu \in \mathbb{R}^P$ follow analogously to Propositions \[prop:solution\_dmu\_eta\] and \[prop:dual\_solution\_dmu\_eta\]. Furthermore, ${\hat{\mathcal{J}}}$ 