Optimization Of The RDF Preconditioners

Saddle point problems arise in many important scientific research fields,for example computational fluid dynamics,approximation theory,domain decomposition methods.Hence it is of great importance for computational science and engineering to investigate the numerical methods for the saddle point problems.In this thesis,we consider the optimization of Relaxed Dimensional Factorization(RDF)preconditioner that is designed for solving the saddle point problems.The studies consist of the optimal choices of the relaxation parameter in the RDF method for solving saddle point problems that are obtained by discretizing the Stokes and Oseen problem from nonlinear Navier-Stokes equations.The analysis is perfomed at continuous level.We first obtain the contraction factor,which is a function of the Fourier frequencies,by the technique of Fourier transform,then we determine the optimal relaxation parameter by solving a minimization problem of the contraction factor.The theoretical analysis shows that the RDF method is independent of the mesh size and the viscosity when applied to 2D or 3D Stokes equations.When the RDF method is applied to the Oseen problem,it depends on both the mesh size and the viscosity.We at last validate our theoretical findings by numerical experiments.