Semi-discrete a priori error analysis for the optimal control of the unsteady Navier-Stokes equations with variational multiscale stabilization. (English) Zbl 1410.49021
Summary: In this work, the optimal control problems of the unsteady Navier-Stokes equations with variational multiscale stabilization (VMS) are considered. At first, the first order continuous optimality conditions are obtained. Since the adjoint equation of the Navier-Stokes problem is a convection diffusion type system, then the same stabilization is applied to it. Semi discrete a priori error estimates are obtained for the state, adjoint state and control variables. Crank-Nicholson time discretization is used to get the fully discrete scheme. Numerical examples verify the theoretical findings and show the efficiency of the stabilization for higher Reynolds number.
MSC:
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 49M25 | Discrete approximations in optimal control |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 76D55 | Flow control and optimization for incompressible viscous fluids |
Keywords:
optimal control; variational multiscale methods; stabilized FEM; unsteady Navier-Stokes; error estimateSoftware:
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