Some new results for solving linear systems arising from computational fluid dynamics problems

In this thesis, we consider the numerical solution of four kinds of linear systems: saddle-point problems, Stokes problems, symmetric systems (positive definite or indefinite), and unsymmetric systems. These systems are related, and all of them arise from the numerical solution of partial differential equations. For saddle-point problems, we introduce a class of expansion methods based on a new solution representation of the general problem. Many difficult computations involved in Uzawa and projection type methods for saddle-point problems are avoided in our approach. For the Stokes problems, by introducing a new variable, we split the linear system into several smaller systems according to its sparse structure. This new variable is then updated so that the split systems eventually produce the solution of the original problem. For symmetric systems that are indefinite and do not have a special sparse structure like saddle-point problems, we propose a class of nested iterative methods to handle them. We study how the convergence rate of the outer iteration is related to the convergence rate of the inner iterations. For symmetric positive definite systems, we propose a class of nested preconditioners and analyze properties of these preconditioners. Some necessary and sufficient conditions for the optimality of these nested preconditioners are established. Finally, for unsymmetric systems, we generalize the concept of spectral equivalence for symmetric positive definite systems to these general systems. We also study some properties of spectrally equivalent matrices and consider their applications for constructing efficient iterative methods.U of I OnlyETDs are only available to UIUC Users without author permissio