In this work, we propose fully nonconforming, locally exactly divergence-free discretizations based on lowest order Crouziex-Raviart finite element and piecewise constant spaces to study the optimal control of stationary double diffusion model presented in [B\"urger, M\'endez, Ruiz-Baier, SINUM (2019), 57:1318-1343]. The well-posedness of the discrete uncontrolled state and adjoint equations are discussed using discrete lifting and fixed point arguments, and convergence results are derived rigorously under minimal regularity. Building upon our recent work [Tushar, Khan, Mohan arXiv (2023)], we prove the local optimality of a reference control using second-order sufficient optimality condition for the control problem, and use it along with an optimize-then-discretize approach to prove optimal order a priori error estimates for the control, state and adjoint variables upto the regularity of the solution. The optimal control is computed using a primal-dual active set strategy as a semi-smooth Newton method and computational tests validate the predicted error decay rates and illustrate the proposed scheme's applicability to optimal control of thermohaline circulation problems.

翻译：本文基于最低阶Crouziex-Raviart有限元和分片常数空间，提出完全非协调、局部精确无散度的离散格式，用于研究文献[B\"urger, M\'endez, Ruiz-Baier, SINUM (2019), 57:1318-1343]中稳态双扩散模型的最优控制问题。利用离散提升和不动点论证讨论了离散无控制状态方程和伴随方程的适定性，并在最低正则性条件下严格推导了收敛性结果。基于我们近期的工作[Tushar, Khan, Mohan arXiv (2023)]，利用控制问题的二阶充分最优性条件证明了参考控制的局部最优性，并结合先优化后离散的方法，直至解的正则性范围内，建立了控制变量、状态变量和伴随变量的最优阶先验误差估计。采用原始-对偶积极集策略作为半光滑牛顿法计算最优控制，数值实验验证了预测的误差衰减率，并展示了所提方案在热盐环流问题最优控制中的适用性。