The Relaxed Preonditioners For Nonsymmetric Saddle Point Problems

In a variety of scientific and engineering applications, such as computational fluid dynam-ics, constrained optimization and electromagnetism, the research problem will be converted to solving a type of large sparse linear system—saddle point problems. The main solvers for saddle point problems contain the direct methods and the iterative methods. The direct methods will require more computational overhead and storage when dealing with the large sparse systems, since the zero elements will fill in "the blank". The iterative methods remain the key ingredient for saddle point problems, but sometimes it may converge slowly for large systems. A common strategy to overcome the drawback mentioned above is to adopt the preconditioning technology.This thesis is concerned with three kinds of relaxed preconditioners for nonsymmetric sad-dle point problems. These preconditioners are based on the DPSS preconditioner proposed by Pan in 2006. We also display the corresponding theoretical results and numerical experiments. The main work can be summarized as followed:1. The DPSS preconditioner is an efficient preconditioner for nonsymmetric saddle point problems. The eigenvalues of the system preconditioned by this preconditioner will converge to the origin or the point (2,0). In this thesis, we propose a VDPSS prconditioner by deleting the shift term in the first factorization of the difference between the DPSS preconditioner and the origin coefficient matrix. The VDPSS preconditioner is much closer to the origin coefficient matrix, and the preconditioned matrix has eigenvalue 1 of algebraic multiplicity at least n. We theoretically analyze the corresponding Krylov subspace. The numerical experiments illustrate the effectiveness of the VDPSS preconditioner from the eigenvalue distribution, iterative counts and CPU time.2. We propose the second relaxed preconditioner, i.e., RDPSS preconditioner, for the nonsymmetric saddle point problems by directly deleting the shift term in the difference between the DPSS preconditioner and the origin coefficient matrix. We not only prove the preconditioned matrix has eigenvalue 1 of algebraic multiplicity at least n, but also analyze the remaining eigenvalues of the preconditioned matrix. The remaining eigenvalues will converge to 0, as the iterative parament α tends to 0 and +∞. The most important point is that we prove that the corresponding iterative method is convergent unconditionally. Numerical experiments shows that the RDPSS preconditioner is more effective for solving the nonsymmetric saddle point problems, and this preconditioner is not parament-dependent (on α).3. Since we need to balance α and α-1 in the difference matrix between the DPSS precon-ditioner and the origin coefficient matrix, we consider knocking out all the two diagonal shift terms in the difference. Then a new relaxed preconditioner which avoid balancing α and α 1 is proposed in the thesis. We prove the corresponding iterative method converge to the exact solution unconditionally. The optical relaxation parament which ensure the fastest convergence can be obtained theoretically. We also analyze the eigenvector structure which also influence the computational overhead. In the end of the thesis, we adopt an inexpensive approximation of the matrix A to solve the subsystem which involving A-1. Numerical experiments illustrate that the new relaxed preconditioner is effective than its predecessors. Although the approximate method requires more iteration steps, but it also bring about shorter CPU time.