A High-Order Fast Direct Solver for Singular Poisson Equations

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.