In this paper, we discuss a priori error estimates for the finite volume element approximation of optimal control problem governed by Stokes equations. Under some reasonable assumptions, we obtain optimal $L^{2}$ -norm error estimates. The approximate orders for the state, costate, and control variables are $O(h^{2})$ in the sense of $L^{2}$ -norm. Furthermore, we derive $H^{1}$ -norm error estimates for the state and costate variables. Finally, we give some conclusions and future works.

中文翻译：

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斯托克斯最优控制问题的有限体积法误差估计

在本文中，我们讨论了由斯托克斯方程控制的最优控制问题的有限体积元近似的先验误差估计。在一些合理的假设下，我们获得最优的$ L ^ {2} $-范数误差估计。就$ L ^ {2} $ -norm而言，状态，costate和控制变量的近似顺序为$ O（h ^ {2}）$。此外，我们导出了状态变量和costate变量的$ H ^ {1} $-范数误差估计。最后，我们给出一些结论和未来的工作。