A High Order Fast Direct Solver for Singular Poisson Equations
Authors: Y. Zhuang, X.-H. Sun
Date: January, 2001
Venue: Journal of Computational Physics, Vol. 171, pp. 79-94 (2001).
Type: Journal
Abstract
We present a fourth order numerical solution method for the singular Neumann boundary problem of Poisson equations. Such problems arise in the solution process of incompressible Navier-Stokes equations and in the time-harmonic wave propaga- tion in the frequence space with the zero wavenumber. The equation is first discretized with a fourth order modified Collatz difference scheme, producing a singular discrete equation. Then an efficient singular value decomposition (SVD) method modified from a fast Poisson solver is employed to project the discrete singular equation into the orthogonal complement of the null space of the singular matrix. In the comple- ment of the null space, the projected equation is uniquely solvable and its solution is proven to be a solution of the original singular discrete equation when the original equation has a solution. Analytical and experimental results show that this newly proposed singular equation solver is efficient while retaining the accuracy of the high order discretization. 2001 Academic Press Key Words: Poisson equation; Neumann boundary condition; SVD; fast Fourier transform (FFT); high order discretization. ©