TY - JOUR

T1 - A High-Order Fast Direct Solver for Singular Poisson Equations

AU - Zhuang, Yu

AU - Sun, Xian He

N1 - Funding Information:
This research was supported in part by the National Science Foundation (NSF) under NSF Grant CCR-9972251 and by the Office of Naval Research (ONR) under the PET program. We also thank the anonymous referees for their comments and suggestions.

PY - 2001/7/20

Y1 - 2001/7/20

N2 - 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 propagation 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 complement 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.

AB - 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 propagation 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 complement 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.

KW - Neumann boundary condition

KW - Poisson equation

KW - SVD

KW - fast Fourier transform (FFT)

KW - high order discretization

UR - http://www.scopus.com/inward/record.url?scp=0002346969&partnerID=8YFLogxK

U2 - 10.1006/jcph.2001.6771

DO - 10.1006/jcph.2001.6771

M3 - Article

AN - SCOPUS:0002346969

VL - 171

SP - 79

EP - 94

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

IS - 1

ER -