Preconditioning Methods For Discretized Equations Of PDE And Saddle Point Problems

In order to solve the discretized equations of partial differential equations (PDEs)and large sparse saddle point system, in this thesis, some preconditioned methods are ob-tained by using some knowledge, e.g., symmetric positive definite matrix, M-matrix, H-matrix, strictly diagonally dominant matrix and banded matrix, and using some methods,e.g., constraint preconditioned method, Krylov subspace method, incomplete LU factor-ization method, HSS iteration method, PHSS iteration method, AHSS iteration method,Augmented Lagrangian method, multiplicative Schwarz method and sparse approximateinverse preconditioned method, and so on. The main achievements derived in this thesisare listed as follows:In this thesis, based on diagonally compensated reduction method and incompleteLU factorization method, a constraint preconditiner is designed and applied to solve largesparse symmetric positive definite systems and theoretically verified the convergence ofrelated iteration method. In numerical experiments, this type of constraint preconditioneris applied to accelerate the convergence rates of Krylov subspace methods, e.g., PCGiteration method, GMRES iteration method and BiCGSTAB iteration method, and thiskind of preconditioned Krylov subspace method is used to solve Helmholtz equation andpoisson equation. Numerical comparison with existing results by number of iterationsand the solution time further verified the efficiency and accuracy of the new constraintpreconditioner.So as to solve generalized saddle point problem, based on HSS iteration method,PHSS iteration method and AHSS iteration method, in this thesis, PAHSS iterationmethod is proposed. We theoretically proved that the PAHSS iteration method convergesunconditionally to the unique solution of saddle point for any positive iteration parame-ters and studied and discussed in detail the distribution of spectrum of the correspondingpreconditioned matrix. In numerical experiments, applied PAHSS iteration method pre-conditioned Krylov subspace method to solve the generalized saddle point problem. Bynumerical comparison with AHSS iteration method, the efficiency and practicability ofPAHSS preconditioner is further verified.For the sake of solving discretized mixed time-harmonic Maxwell equations, two kinds of preconditioners are presented in this thesis, which respectively augmentedLagrangian-based block triangular preconditioner and multiplicative block precondi-tioner. By augmented Lagrangian method, the augmented Lagrangian-based block tri-angular preconditioner is proposed and employed augmented Lagrangian-based block tri-angular preconditioner to solve sadlle point problem and studied in detail the distributionof spectrum of the corresponding preconditioned matrix. In numerical experiments, thistype of preconditioner is applied to accelerate the convergence rates of Krylov subspacemethod to solve symmetric indefinite saddle point problem and the efficiency is suc-cessfully verified. According to multiplicative block method or multiplicative Schwarzmethod, a new type of multiplicative block preconditioner is established. Used multi-plicative block preconditioner to solve saddle point problem and analyzed and studied indetail the spectral properties of the corresponding preconditioned matrix. In numericalexperiments, multiplicative block preconditioner is applied to accelerate Krylov subspacemethod for discretized mixed time-harmonic Maxwell equations for any wave number k2.From two aspects of number of iterations and the solution time, by numerical comparisonwith exiting block diagonal preconditioner and two classes of block triangular precondi-tioners, the performance of the multiplicative block preconditioner is further verified.In this thesis, the upper bound of the inverse of strictly diagonally dominant penta-diagonal matrix is considered as approximate inverse preconditioner for effectively solv-ing Toeplitz system. Subsequently, the lower and upper bounds of the inverse of strictlydiagonally dominant seventh-diagonal matrix are estimated, when the concerned matrixis a strictly diagonally dominant seventh-diagonal M-matrix. According to theoreticalanalysis, the upper bounds is just the accurate inverse of the original matrix. At the sametime, the efficiency of the bounds is numerically verified.