## Fast iterative solvers for convection-diffusion control problems

John W. Pearson and Andrew J. Wathen

### Abstract

In this manuscript, we describe effective solvers for the optimal control of stabilized convection-diffusion control problems. We employ the Local Projection Stabilization, which results in the same matrix system whether the discretize-then-optimize or optimize-then-discretize approach for this problem is used. We then derive two effective preconditioners for this problem, the first to be used with MINRES and the second to be used with the Bramble-Pasciak Conjugate Gradient method. The key components of both preconditioners are an accurate mass matrix approximation, a good approximation of the Schur complement, and an appropriate multigrid process to enact this latter approximation. We present numerical results to illustrate that these preconditioners result in convergence in a small number of iterations, which is robust with respect to the step-size $h$ and the regularization parameter $\beta$ for a range of problems.

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### Key words

PDE-constrained optimization, convection-diffusion control, preconditioning, Local Projection Stabilization, Schur complement.

### AMS subject classifications

49M25, 65F08, 65F10, 65N30.

### Links to the cited ETNA articles

 [25] Vol. 34 (2008-2009), pp. 125-135 Andy Wathen and Tyrone Rees: Chebyshev semi-iteration in preconditioning for problems including the mass matrix