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

In recent years,in many applications such as engineering and scientific compu-tations,numerical solutions of the optimal control problems constrained by partial differential equations have widely attracted people’s attentions.In this thesis,we consider the numerical solutions of the time-independent Navier-Stokes optimal con-trol problems.By Q2-Q1 mixed finite element discretization,a large sparse and ill-conditioned saddle point problem is obtained.Solving this kind of problems ef-ficiently is the key to solve the control problems,the main work of this thesis is to construct splitting preconditioners for solving the large saddle point problems efficiently by the preconditioned Krylov subspace method.A class of splitting preconditioners for the general saddle point problems are first proposed.In theory,the spectral properties of the corresponding precondi-tioned matrix are discussed and the unconditional convergence theorem is given.Then these results are generalized for the generalized saddle point problems.For a matrix with a double saddle point structure discretized from the optimal control problems,we propose a class of splitting preconditioners.The eigenvalues and eigen-vectors of the preconditioned system have been discussed.Finally,the efficiency of preconditioners is illustrated through numerical experiments.The contributions of this thesis include:(1)Based on the extended shift-splitting preconditioner which was presented by Lu et al.in JCAM in 2017,we generalize the conclusions in the literature and prove the unconditional convergence theorem of the saddle point problems.In addition,the spectral properties of the preconditioned matrix are investigated.(2)Splitting preconditioners are constructed for the generalized saddle point problems.The corresponding convergence theorem and the spectral properties of the preconditioned matrix are discussed.In addition,the efficiency of these precon-ditioners are compared by the numerical experiments.(3)Based on the time-independent Navier-Stokes optimal control problems,the splitting preconditioners for the matrix with a double saddle point structure are proposed and the corresponding convergence theorem is also given.The efficiency of the preconditioned GMRES method is illustrated by approximate implementation of our preconditioners.