This study introduces the divergence-conforming discontinuous Galerkin finite element method (DGFEM) for numerically approximating optimal control problems with distributed constraints, specifically those governed by stationary generalized Oseen equations. We provide optimal a priori error estimates in energy norms for such problems using the divergence-conforming DGFEM approach. Moreover, we thoroughly analyze $L^2$ error estimates for scenarios dominated by diffusion and convection. Additionally, we establish the new reliable and efficient a posteriori error estimators for the optimal control of the Oseen equation with variable viscosity. Theoretical findings are validated through numerical experiments conducted in both two and three dimensions.

翻译：本文引入散度相容间断Galerkin有限元方法（DGFEM），用于数值逼近具有分布约束的最优控制问题，特别针对定常广义Oseen方程描述的问题。我们利用散度相容DGFEM方法，给出了此类问题在能量范数下的最优先验误差估计。此外，详细分析了扩散主导和对流主导情形下的$L^2$误差估计。同时，针对变粘度Oseen方程的最优控制问题，建立了新型可靠且高效的后验误差估计子。通过二维和三维数值实验验证了理论结果。