Functional Analysis And Convergence Study Of The Krylov Subspace Method

Many scientific and engineering computational model problems,after discretization by methods such as finite difference,finite element or moment method,eventually require solving a linear system of one or a series of large sparse matrices.At present,the Krylov subspace method is the preferred method for solving large-scale sparse linear equations.The research of Krylov subspace method in the context of functional analysis is a new thinking of analyzing and studying iterative methods.The finite element operator is a constraint on the operator of infinite-dimensional Hilbert space in its finite-dimensional subspace,which ensures the rationality of the research in the context of Hilbert space.Many researchers have analyzed the iterative method in the background of Hilbert space and other possible infinite-dimensional abstract spaces,and have obtained very good results.Bank and Dupont studied the finite difference technique in the context of an abstract framework suitable for finite elements.Kirby studied the finite element discrete condition numbers,preconditioning techniques,and iterative methods of partial differential equations in the context of functional analysis.This paper draws on the new perspectives of studying the iterative method and preconditioning technology from the perspective of functional analysis.It mainly involves the functional analysis of the Krylov subspace method for large-scale linear systems.Firstly,in the context of functional analysis,the focus is on the functional analysis of the preconditioned generalized CG method in the Hilbert space,focusing on the consistency between the full version of the generalized CG algorithm and the truncated version with the zero control term.The coefficient operator is preconditioned by the symmetry part of the coefficient operator,and the consistency between the preconditioned full version generalized CG algorithm and the truncated version algorithm with zero control term is proved.Secondly,as a practical application of the iterative method theory of functional analysis,taking the elliptic equation as an example,the existence and uniqueness of the solution of the elliptic equation are proved from the perspective of continuity and coercivity of the Lax-Milgram lemma.And preconditioned generalized CG Method is applied to perform related analysis in Hilbert Space.The consistency between the algorithm of the full version and the truncated version with zero control term is proved,and the relevant condition number estimation is given.Finally,the boundary value problems of one-dimension and two-dimension convective diffusion equations are studied from the perspective of numerical experiments.The graphical representation of the numerical solution of the equation is given.The convergence rates of truncated version of the generalized CG method and preconditioned generalized CG method are compared,and the superiority of the truncated version preconditioned generalized CG method is proved.