Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-27T23:02:25.892Z Has data issue: false hasContentIssue false
  • English
  • Français

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:[email protected] (M. M. Butt), [email protected] (Y. Yuan)
*Corresponding author. Email addresses:[email protected] (M. M. Butt), [email protected] (Y. Yuan)
Get access

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Bacuta, C., Vassilevski, P. and Zhang, S., A new approach for solving Stokes systems arising from a distributive relaxation method, Numer. Methods Partial Differ. Equ., 27 (2011), pp. 898–914.Google Scholar
[2] Benzi, M., G. H., , Golub, and Liesen, J., Numerical solution of saddle point problems, Acta Numerica, 14 (2005), pp. 1–137.CrossRefGoogle Scholar
[3] Borzì, A. and Schulz, V., Multigrid methods for PDE optimization, SIAM Review, 51 (2009), pp. 361–395.Google Scholar
[4] Brandt, A., Multi-level adaptive solutions to boundary-value problems, Math. Comp. 31 (1977), pp. 333–390.Google Scholar
[5] Brandt, A. and Dinar, N., Multigrid solutions to elliptic flow problems, in Parter, S. V. (ed.), Numerical Methods for PDEs, Academic Press, New York, 1979, pp. 53–147.Google Scholar
[6] Butt, M. M., A multigrid solver for stokes control problems, Int. J. Comput. Math.Google Scholar
[7] Butt, M. M. and Borzì, A., Formulation and multigrid solution of Cauchy-Riemann optimal control problems, Computing and Visualization in Science, 14 (2011), pp. 79–90.Google Scholar
[8] Dendy, J. E. Jr and Moulton, J. D., Black Box Multigrid with coarsening by a factor of three, Numer. Linear Algebra Appl., 17 (2010), pp. 577–598.Google Scholar
[9] Drgnescu, A. and Soane, A. M., Multigrid solution of a distributed optimal control problem constrained by the Stokes equations, Appl. Math. Comput., 219 (2013), pp. 5622–5634.Google Scholar
[10] Evans, L.C., Partial Differential Equations, Graduate Studies in Mathematics, American Mathematical Society, 128(1-2) (2002), pp. 55–82.Google Scholar
[11] Girault, V. and Raviart, P. A., Finite Element Methods for NavierâĂŞStokes Equations: Theory and Algorithms, Springer Series in Computational Mathematics, Springer-Verlag, Berlin, 5 (1986).Google Scholar
[12] Kellogg, R. B. and Osborn, J. E., A regularity result for the Stokes problem in a convex polygon, J. Funct. Anal., 21 (1976), pp. 397–431.Google Scholar
[13] Hackbusch, W., Multi-grid Methods and Applications, Springer-Verlag, 960(4) (1985), pp. 558–575.Google Scholar
[14] Hackbusch, W., Elliptic Differential Equations, Springer, New York, 1992.Google Scholar
[15] Ito, K. and Kunisch, K., Lagrange Multiplier Approach to Variational Problems and Applications, SIAM, 2008.Google Scholar
[16] Kollmann, M. and Zulehner, W., A Robust Preconditioner for Distributed Optimal Control for Stokes Flow with Control Constraints, In Cangiani, Andrea, Davidchack, Ruslan L., Numerical Mathematics and Advanced Applications 2011, Springer Berlin Heidelberg, (2013), pp. 771–779.Google Scholar
[17] Lions, J. L., Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971.Google Scholar
[18] Oosterlee, C. W. and Gaspar, F. J., Multigrid methods for the stokes system, Comput. Sci. Eng., 8 (2006), pp. 34–43.CrossRefGoogle Scholar
[19] Takacs, S., A robust all-at-once multigrid method for the Stokes control problem, Numerische Mathematik, 130 (2015), pp. 517–540 Google Scholar
[20] Tröltzsch, F., Optimal control of partial differential equations: theory, methods and applications, AMS, 2010.Google Scholar
[21] Trottenberg, U., Oosterlee, C. and Schüller, A., Multigrid, Academic Press, London, 2001.Google Scholar
[22] Wang, M. and Chen, L., Multigrid methods for the stokes equations using distributive Gauss–seidel relaxations based on the least squares commutator, J. Sci. Comput. 56 (2013), pp. 409–431.CrossRefGoogle Scholar
[23] Wittum, G., Multi-grid methods for Stokes and Navier-Stokes equations, Numerische Mathematik, 54 (1989), pp. 543–563.Google Scholar