The Numerical Methods For Some Special Sparse Linear Systems

Large scale sparse linear systems arise in many area, such as fluid dynamics,structural analysis, numerical calculation of electromagnetic fields, and so on.When discreting the PDEs which describe the phenomena, we generally obtainsparse linear systems. Therefore, Solving these systems efficiently is of vitalimportance for solving the whole problem. So, study on the numerical methods forsolving large sparse linear systems is an important field in large scale scientific andEngineering computation. Furthermore, many large scale linear systems arisingin practical problems are often of some special forms or particular structures,Therefore, this thesis investigates the fast and efficiently numerical method forsolving some special linear systems, the text is divided into five chapters.Chapter 1 gives an introduction of the origin, history, state of iterative methodsfor solving large sparse linear system of equations. And we also briefly introducethe special linear systems which were studied in this paper.In chapter 2, a novel restarted GMRES method for solving large sparseshifted linear systems is developed. Restarting is carried out by augmentingthe Krylov subspaces with some error approximations generated by the seedsystem, we firstly seek the solution of the seed system in the augmented Krylovsubspaces and acquire the solutions of add systems by making the residual vectorsparallel to the residual vector of seed system. The new method preserves the niceproperty that allows solving the seed and add systems in one subspace. And italso effectively accelerates the convergence of the restarted GMRES method forsolving the shifted linear systems. Numerical experiments indicates the efficiencyof the new method.In Chapter 3, we considers the block-tridiagonal linear system of equations,a variant of tangential filtering preconditioners is proposed. The new variant isbased on a twisted block factorization along with certain filtering property. Inthe practical application, a class of composite preconditioners are tested, whichare constructed by combining the twisted tangential filtering decomposition preconditionerwith the classical ILU(O) preconditioner in a nmltiplicative way. The performance of the new preconditioners are compared with other classical preconditioners,the superiority and the weakness of the preconditioners are proposed.In chapter 4, we illustrate the preconditioned gradient based iterative methodwhich can be derived by reasonable choice of two auxiliary matrices. The strategyis a natural generalization of the splitting iteration methods for linear systemsof equations. The performance of the preconditioned gradient based iterativemethod is compared with the original method on several numerical examples. Abetter convergence behavior is revealed, and the influence of an step-size parameteris experimentally studied.In the last chapter, we propose a preconditioner for a class of the generalizedsaddle point problems. The preconditioner is based on matrix splitting, and anew proposed two parameters splitting iteration technique [Z. Z Bai and G. H.Golub, IMA J. Numer. Anal.,27, (2007), pp.1-23]. The spectral properties of thepreconditioned matrix are discussed in detail. Numerical experiments are givento show the conclusion and the efficiency of the precondtioner.