Volume 44, pp. 53-72, 2015.

On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems

John W. Pearson

Abstract

The development of preconditioners for PDE-constrained optimization problems is a field of numerical analysis which has recently generated much interest. One class of problems which has been investigated in particular is that of Stokes control problems, that is, the problem of minimizing a functional with the Stokes (or Navier-Stokes) equations as constraints. In this manuscript, we present an approach for preconditioning Stokes control problems using preconditioners for the Poisson control problem and, crucially, the application of a commutator argument. This methodology leads to two block diagonal preconditioners for the problem, one of which was previously derived by W. Zulehner in 2011 [SIAM J. Matrix Anal. Appl., 32 (2011), pp. 536–560] using a nonstandard norm argument for this saddle point problem, and the other of which we believe to be new. We also derive two related block triangular preconditioners using the same methodology and present numerical results to demonstrate the performance of the four preconditioners in practice.

Full Text (PDF) [437 KB]

Key words

PDE-constrained optimization, Stokes control, saddle point system, preconditioning, Schur complement, commutator

AMS subject classifications

65F08, 65F10, 65F50, 76D07, 76D55, 93C20

Links to the cited ETNA articles

[7]Vol. 35 (2009), pp. 257-280 Howard C. Elman and Ray S. Tuminaro: Boundary conditions in approximate commutator preconditioners for the Navier-Stokes equations
[17]Vol. 40 (2013), pp. 294-310 John W. Pearson and Andrew J. Wathen: Fast iterative solvers for convection-diffusion control problems
[25]Vol. 34 (2008-2009), pp. 125-135 Andy Wathen and Tyrone Rees: Chebyshev semi-iteration in preconditioning for problems including the mass matrix

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