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Convergence Analysis and Error Estimate for Distributed Optimal Control Problems Governed by Stokes Equations with Velocity-Constraint

Convergence Analysis and Error Estimate for Distributed Optimal Control Problems Governed by Stokes Equations with Velocity-Constraint
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Year:    2022

Author:    Liang Ge, Haifeng Niu, Jianwei Zhou

Advances in Applied Mathematics and Mechanics, Vol. 14 (2022), Iss. 1 : pp. 33–55

Abstract

In this paper, 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. Optimality conditions of the continuous and discretized systems are deduced with the Karush-Kuhn-Tucker conditions and a Lagrange multiplier depending on the constraint. To solve the equivalent systems with high accuracy, Galerkin spectral approximations are employed to discretize the constrained optimal control systems. Meanwhile, we adopt a parameter $\lambda$ in the pressure approximation space, which also guarantees the inf-sup condition, and study a priori error estimates for the velocity and pressure. Specially, an efficient algorithm based on the Uzawa algorithm is proposed and its convergence results are investigated with rigorous analyses. Numerical experiments are performed to confirm the theoretical results.

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Journal Article Details

Publisher Name:    Global Science Press

Language:    English

DOI:    https://doi.org/10.4208/aamm.OA-2020-0302

Advances in Applied Mathematics and Mechanics, Vol. 14 (2022), Iss. 1 : pp. 33–55

Published online:    2022-01

AMS Subject Headings:    Global Science Press

Copyright:    COPYRIGHT: © Global Science Press

Pages:    23

Keywords:    Optimal control spectral approximation Stokes equation convergence analysis.

Author Details

Liang Ge

Haifeng Niu

Jianwei Zhou

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