A Class Of Splitting Preconditioned Iterative Methods For Solving Optimal Control Problems Of Navier-Stokes Equations

In this thesis,the numerical methods of the time-independent Navier-Stokes op-timal control problems are discussed.By Q2-Q1 mixed finite element discretization,this kind of control problem is transformed into a large and sparse nonsymmetric saddle point problem.In order to solve this kind of ill-conditioned saddle point prob-lems efficiently,a class of EPGSS(extended parameterized generalized shift-splitting)preconditioner is proposed to preprocess Krylov subspace method.For the general nonsymmetric saddle point problems(the(2,2)block matrix is a zero matrix),the corresponding EPGSS iteration method can be applied not only to the nonsingular saddle point problems but also to the singular ones.In theory,the convergence and semi-convergence of EPGSS iterative method are analyzed.What's more,the eigenvalues and the eigenvectors of the preconditioned matrix are discussed.In addition,we apply the splitting preconditioner to the generalized nonsymmetric saddle point problems(the(2,2)block matrix is a symmetric positive definite ma-trix)and give the corresponding convergence analysis.Finally,the effectiveness of the EPGSS preconditioner and the corresponding iterative method are illustrated by numerical experiments.The contributions of this thesis include:(1)Based on the shift-splitting preconditioner presented by Huang et alon Com-put.Math.Appl.in 2018,aclass of new splitting preconditioners are constructed for the GMRES method of the general nonsymmetric saddle point problems(the(2,2)block matrix is a zero matrix),which extend the original preconditioners.Theoretical analysis and numerical experiments are carried out,which show the effectiveness of the new preconditioned GMRES method.(2)The extended preconditioners are applied to the GMRES method for the gen-eralized nonsymmetric saddle point problems(the(2,2)block matrix is a symmetric positive definite matrix)and theoretical analysis and numerical experiments are car-ried out to prove the effectiveness of the preconditioned GMRES method.