This work examines the distributed optimal control of generalized Oseen equations with non-constant viscosity. We propose and analyze a new conforming augmented mixed finite element method and a Discontinuous Galerkin (DG) method for the velocity-vorticity-pressure formulation. The continuous formulation, which incorporates least-squares terms from both the constitutive equation and the incompressibility condition, is well-posed under certain assumptions on the viscosity parameter. The CG method is divergence-conforming and suits any Stokes inf-sup stable velocity-pressure finite element pair, while a generic discrete space approximates vorticity. The DG scheme employs a stabilization technique, and a piecewise constant discretization estimates the control variable. We establish optimal a priori and residual-based a posteriori error estimates for the proposed schemes. Finally, we provide numerical experiments to showcase the method's performance and effectiveness.

翻译：本文研究具有非常数粘度的广义Oseen方程分布最优控制问题。针对速度-涡度-压力公式，我们提出并分析了一种新的协调增强混合有限元方法和一种间断伽辽金（DG）方法。该连续公式结合了本构方程和不可压缩条件的最小二乘项，在粘度参数满足特定假设条件下是适定的。协调有限元方法具有散度协调特性，适用于任意满足Stokes inf-sup稳定条件的速度-压力有限元对，同时采用通用离散空间逼近涡度。DG方案采用稳定化技术，并通过分段常数离散化估计控制变量。我们为所提方案建立了最优先验误差估计和基于残差的后验误差估计。最后，通过数值实验验证了该方法的性能与有效性。