Splitting Preconditioned Iterative Methods For The Optimal Control Problem Constrained By The Navier-Stokes Equations

In recent years, optimal control problems with PDE constrains receive more and more attention in the field of the applied science and the scientific comput-ing. After a discretization, this kind of problem leads to a generalized saddle point system. However, because of the ill-condition of the coefficient matrix, the con-vergence speed is fairly slow, when the Krylov subspace methods are used directly. To improve the convergence speed, we will adopt preconditioned Krylov subspace methods. Therefore, finding the effective preconditioners is our main work.We discuss the time-independent Navier-Stokes optimal control problem in this thesis. After Q2-Q1 mixed finite element discretization, KKT conditions and some permutations, we get a generalized nonsymmetric saddle point system whose (1,1) block of the coefficient matrix has a positive definite symmetric part. For the non-symmetric saddle point system, based on two different splittings of the coefficient matrix, we give three preconditioners PS, PRS and PGRS. Then, we investigate the spectral properties of the corresponding preconditioned matrix. Finally, some numerical experiments are presented to illustrate the effectiveness of the three pre-conditioners mentioned above.The contributions of this thesis include:(1) For the saddle point system from the discretization of the time-independent Navier-Stokes optimal control problem, based on the idea in [31], a new splitting preconditioner PS and its relaxed variant PRS are proposed. Then, the theoretical analysis and numerical experiments are given.(2) Based on a new splitting of the coefficient matrix A, a generalized relaxed splitting preconditioner PGRS is proposed and its effectiveness is illustrated by some numerical experiments.