In two and three dimensional Lipschitz, but not necessarily convex, polytopal domains, we propose and analyze a posteriori error estimators for an optimal control problem involving the stationary Navier--Stokes equations; control constraints are also considered. We devise two strategies of discretization: a semi discrete scheme where the control variable is not discretized -- the so-called variational discrezation approach -- and a fully discrete scheme where the control is discretized with piecewise quadratic functions. For each solution solution technique, we design an a posteriori error estimator that can be decomposed as the sum of contributions related to the discretization of the state and adjoint equations and, additionally, the discretization of the control variable for when the fully discrete scheme is considered. We prove that the devised error estimators are reliable and also explore local efficiency estimates. Numerical experiments reveal a competitive performance of adaptive loops based on the devised a posteriori error estimators.
翻译:在二维和三维的Lipschitz, 但不一定是相交的多方形域中, 我们提议和分析一个后端误差估计器, 以优化控制问题, 包括固定的 Navier- Stokes 方程式; 也考虑控制限制。 我们设计了两种离散策略: 一个半离散策略, 控制变量没有分离 -- 所谓的变异解法 -- 和一个完全独立的计划, 控制与片断的二次函数分离。 对于每一种解决方案技术, 我们设计了一个后端误差估计器, 可以分离为与状态离散和连接方程式有关的贡献总和, 以及在审议完全离散的方程式时控制变量的离散。 我们证明, 设计的误差估计器是可靠的, 并探索本地效率估计。 数字实验显示,根据设计后方误判测器, 适应循环的竞争性表现。