Preconditioning Techniques For Saddle Point Problems With Application In Navier-Stokes Equations

Saddle point problems is a kind of important system of linear equations, derived from many fields of science and engineering applications, including computational fluid dynamics, constrained optimization, mixed finite element of elliptic PDEs, generalized least-squares problems. The object of this thesis is stable Navier-Stokes equations by the finite element discrete linear equations produced the saddle point problems. The way to solving the linear system can be classified into direct method and iterative method. De-pending on a large number of matrix order, the direct method required more storage ca-pacity and computing time than the Krylov subspace iteration method to solve the problem of the saddle point. So the advantage of Krylov subspace iteration method solving is obvious, which need a small amount of calculation and little memory con-sumption. But in fact, as more iteration number is required, it is necessary that the data was calculated by preconditioning technology to speed up the convergence rate of the Krylov subspace iteration method.Based on the character of finite element discrete stability of Navier-Stokes equa-tions, the saddle point problem with special structure is obtained. According to special structure, A relaxed splitting preconditioner based on matrix splitting is introduced. Then combinated the relative spectral distributional properties of the preconditioned matrix and Krylov subspace methods with optimal parameters to show that the precon-ditioned GMRES is valid.Relying on saddle point problems of Navier-Stokes equations with stability finite element discrete and taking advantage of augmented Lagrangian preconditioner, we come up with the incomplete augmented Lagrangian preconditioner. Through analysis-ing the spectral properties of preconditioned matrix, found that all of the eigenvalues of preconditioned matrix are distributed on the right half-plane. There are at least n ei-genvalues (recall that n is the number of velocity degress of freedom) which value1and belong to the area[0,1]×(-1/2,1/2). Experiment find the preconditioned GMRES itera-tion number as the problem size increases with the increased slightly. Iteration number does not depend on the grid size and coefficient of viscosity, which are same for uni- form grid and stretch mesh. While the optimal parameters has nothing to do with the grid size, it sightly dependent on kinematic viscosity coefficient. Experimental optimal parameter decreases with the decrease of viscosity coefficient. But when the viscosity coefficient is invariant, the optimal experimental parameters as the problem size is not change and change. Through the numerical experiment shows that numerical results of eigenvalues distribution and theory analysis are consistent...