A Full Multigrid Method for Distributed Control Problems Constrained by Stokes Equations. Issue 3 (20th June 2017)

• Record Type: Journal Article
• Authors: Butt, M. M.; Yuan, Y.
• Is Part Of: Numerical mathematics. Volume 10:Issue 3(2017)
• Page Start: 639
• Page End: 655
• Publication Date: 2017-06-20
• Subjects: 35Q93, -- 65N55, -- 49K20; PDE-constrained optimization, -- stokes equations, -- multigrid, -- staggered grids, -- finite difference; Numerical analysis -- Periodicals; Numerical analysis; Periodicals; 518.05
• DOI: 10.4208/nmtma.2017.m1637
• Languages: English
• ISSNs: 1004-8979

Abstract

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.