Restrictive Preconditioning for Convection-Diffusion Distributed Control Problems

The restrictive preconditioning technique is employed in the preconditioned conjugate gradient and preconditioned Chebyshev iteration methods for the saddle point linear systems arising in convection-diffusion control problems. Utilizing an appropriate approximation of Schur complement, one obtains preconditioned matrix with eigenvalues located in the interval [1/2, 1]. The convergence rate of the methods are studied. Unlike the restrictively preconditioned conjugate gradient method, the restrictively preconditioned Chebyshev iteration method is more tolerant to the inexact execution of the preconditioning. It indicates that the preconditioned Chebyshev iteration method is more practical when dealing with large scale linear systems. Theoretical and numerical results demonstrate that the iteration count of the solvers used do not depend on the mesh size, the regularization parameter and on the Peclet number.