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A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations

Part of: Equations of mathematical physics and other areas of application Partial differential equations, boundary value problems Optimality conditions

Published online by Cambridge University Press:  20 June 2017

M. M. Butt  and
Y. Yuan
M. M. Butt*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing 100080, China; and Higher Education Department, Government of the Punjab, Lahore 54000, Pakistan
Y. Yuan*
Affiliation:
Institute of Computational Mathematics and Scientific/Engineering Computing, Chinese Academy of Sciences, Beijing 100080, China
*
*Corresponding author. Email addresses:mmunirbutt@gmail.com (M. M. Butt), yyx@lsec.cc.ac.cn (Y. Yuan)
*Corresponding author. Email addresses:mmunirbutt@gmail.com (M. M. Butt), yyx@lsec.cc.ac.cn (Y. Yuan)
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Abstract

A full multigrid method with coarsening by a factor-of-three to distributed control problems constrained by Stokes equations is presented. An optimal control problem with cost functional of velocity and/or pressure tracking-type is considered with Dirichlet boundary conditions. The optimality system that results from a Lagrange multiplier framework, form a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods with finite difference discretization on staggered grids. A coarsening by a factor-of-three is used on staggered grids that results nested hierarchy of staggered grids and simplified the inter-grid transfer operators. A distributive-Gauss-Seidel smoothing scheme is employed to update the state- and adjoint-variables and a gradient update step is used to update the control variables. Numerical experiments are presented to demonstrate the effectiveness and efficiency of the proposed multigrid framework to tracking-type optimal control problems.

Keywords

PDE-constrained optimizationstokes equationsmultigridstaggered gridsfinite difference

MSC classification

Secondary: 35Q93: PDEs in connection with control and optimization 65N55: Multigrid methods; domain decomposition 49K20: Problems involving partial differential equations
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 639 - 655
Copyright
Copyright © Global-Science Press 2017 

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