Fast iterative solvers for convection-diffusion control problems

In this manuscript, we describe effective solvers for the optimal control of stabilized convectiondiffusion

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 ? for

a range of problems.