Numerical Algorithms

, Volume 74, Issue 1, pp 19–37

Comparison of preconditioned Krylov subspace iteration methods for PDE-constrained optimization problems

Stokes control
Original Paper

DOI: 10.1007/s11075-016-0136-5

Cite this article as:
Axelsson, O., Farouq, S. & Neytcheva, M. Numer Algor (2017) 74: 19. doi:10.1007/s11075-016-0136-5


The governing dynamics of fluid flow is stated as a system of partial differential equations referred to as the Navier-Stokes system. In industrial and scientific applications, fluid flow control becomes an optimization problem where the governing partial differential equations of the fluid flow are stated as constraints. When discretized, the optimal control of the Navier-Stokes equations leads to large sparse saddle point systems in two levels. In this paper, we consider distributed optimal control for the Stokes system and test the particular case when the arising linear system can be compressed after eliminating the control function. In that case, a system arises in a form which enables the application of an efficient block matrix preconditioner that previously has been applied to solve complex-valued systems in real arithmetic. Under certain conditions, the condition number of the so preconditioned matrix is bounded by 2. The numerical and computational efficiency of the method in terms of number of iterations and execution time is favorably compared with other published methods.


PDE-constrained optimization problems Finite elements Iterative solution methods Preconditioning 

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Institute of Geonics AS CROstravaCzech Republic
  2. 2.Department of Information TechnologyUppsala UniversityUppsalaSweden

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