Null-space Preconditioned Subspace Methods For Solving Navier-Stokes Optimal Control Problems

This thesis devotes to the numerical methods for solving the time-independent Navier-Stokes optimal control problem. The mixed finite element discretization of this class of problems leads to a large and sparse saddle point problem, while the coefficient matrix of this saddle point problem is often ill-conditioned. In recent years, more and more attentions were paid to how to efficiently solve this problem.In this thesis, we study the effective solution method for this saddle point problem by the Krylov subspace methods with Null-space preconditioners and approximate preconditioners respectively.In order to solve this saddle point problem by the preconditioned Krylov sub-space methods efficiently, firstly, we introduce a class of Null-space preconditioners based on the Null-space factorization of the coefficient matrix of the saddle point problem, which are P1 and P2. The spectral properties of the preconditioned sys-tem are analyzed. Then we further proposed approximate preconditioners P3 and P4 based on the incomplete LU factorization of the different sub-blocks of the coef-ficient matrix. The eigenvalues and eigenvectors of the preconditioned system have been analyzed. Finally, several numerical experiments are given to illustrate the efficiency of these preconditioners.The contributions of this thesis include:(1)Based on the Null-space factorization of the coefficient matrix, two Null-space preconditioners P1 and P2 are presented. Theory analysis of the spectral properties and the numerical results of the preconditioned matrix are given.(2)By using incomplete LU factorization of the different sub-blocks of the co-efficient matrix, two approximate preconditioners P3 and P4 are presented and the effectiveness of these preconditioners are compared by the numerical experiments.(3)The effectiveness of the preconditioned Krylov subspace methods with dif-ferent preconditioners presented in this thesis are compared by the numerical ex-periments in different point of view, some analyses and conclusions of the numerical results are given.