On order-reducible sinc discretizations and block-diagonal preconditioning methods for linear third-order ordinary differential equations

Abstract

By introducing a variable substitution we transform the two-point boundary value problem of a third-order ordinary differential equation into a system of two second-order ordinary differential equations. We discretize this order-reduced system of ordinary differential equations by both sinc-collocation and sinc-Galerkin methods, and average these two discretized linear systems to obtain the target system of linear equations. We prove that the discrete solution resulting from the linear system converges exponentially to the true solution of the order-reduced system of ordinary differential equations. The coefficient matrix of the linear system is of block two-by-two structure and each of its blocks is a combination of Toeplitz * Supported by The Hundred Talent and diagonal matrices. Because of its algebraic properties and matrix structures, the linear system can be effectively solved by Krylov subspace iteration methods such as GMRES preconditioned by block-diagonal matrices. We demonstrate that the eigenvalues of certain approximation to the preconditioned matrix are uniformly bounded within a rectangle on the complex plane independent of the size of the discretized linear system, and we use numerical examples to illustrate the feasibility and effectiveness of this new approach.

On the Shift-HSS Splitting Method for Nonsingular Saddle Point Problem

“…In recent year, a large amount of contributions have been put into developing efficient iteration algorithms and preconditioning techniques for solving the saddle point system (1) with different structures, such as HSS-type methods [1,2,3], block preconditioning methods [4], Uzawa-type methods [5], SOR-like method [6], constraint preconditioning methods [7], block and approximate Schur complement preconditioners [8] and (generalized) shift-splitting iteration methods [9][10][11][12], and so on.…”

The Explicit Inverses of CUPL-Toeplitz and CUPL-Hankel Matrices

“…As is well known, Toeplitz matrices family are also structured matrices family and have important applications in various disciplines including the elliptic Dirichlet-periodic boundary value problems [1], sinc discretizations of partial and ordinary differential equations [2][3][4][5][6][7], signal processing [8], numerical analysis [8], system theory [8], etc.…”