A high-order fast direct solver for singular Poisson equations

We present a fourth order numerical solution method for the singular Neuman n boundary problem of Poisson equations. Such problems arise in the solutio n process of incompressible Navier-Stokes equations and in the time-harmoni c wave propagation in the frequence space with the zero wavenumber. The equ ation 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 emp loyed to project the discrete singular equation into the orthogonal complem ent 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 pro ven to be a solution of the original singular discrete equation when the or iginal equation has a solution. Analytical and experimental results show th at this newly proposed singular equation solver is efficient while retainin g the accuracy of the high order discretization. (C) 2001 Academic Press.