A multigrid solver for Stokes control problems

ABSTRACT

Multigrid solvers for distributed optimal control problems constrained by Stokes equations are presented. The distributed velocity tracking problem is considered with Dirichlet boundary conditions. The optimality system of the control problem that results from a Lagrange multiplier framework, forms a linear system connecting the state, adjoint, and control variables. We investigate multigrid methods on staggered grids. A coarsening by a factor of three is introduced that results in a nested hierarchy of staggered grids and simplified the intergrid transfer operators. On these grids a distributive Gauss–Seidel smoothing scheme is employed. Numerical experiments are performed to validate the effectiveness and efficiency of the proposed multigrid staggered grid framework.

KEYWORDS:

• Stokes equations
• PDE-constrained optimization
• multigrid
• staggered grids
• control-constrained

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• 35Q93
• 49K20
• 65N55

Acknowledgements

M. M. Butt thank to Prof Ya-xiang Yuan for the constructive comments and suggestions during this work. The author also thank the anonymous referees for their careful reading and valuable remarks that helped to improve the paper significantly.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by Chinese Academy of Sciences President's International Fellowship Initiative [PIFI2015].