| Abstract: |
| In this work, spectral approximations for distributed optimal control problems governed by the Stokes equation are considered. And the constraint set on velocity is stated with $L^2$-norm. The optimality conditions of the continuous and discretized systems are deduced with the Karush-Kuhn-Tucker conditions. The Lagrange multiplier is dependent on the constraint. Galerkin spectral approximations are employed to solve the optimal control systems. Meanwhile, adopted the first inf-sup condition, we study {\it a priori} error estimates for the velocity and pressure. Specially, an efficient algorithm based on the Uzawa algorithm is proposed and its convergence analysis is proved in detail. Numerical experiments are performed to confirm the theoretical results. |
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