Fast direct solver for the biharmonic equation on a disk and its application to incompressible flows

We develop a simple and efficient FFT-based fast direct solver for the biharmonic equation on a disk. The biharmonic equation is split into a coupled system of harmonic problems. We first use the truncated Fourier series expansion to derive a set of coupled singular ODEs, then we solve those singular equations by second-order finite difference discretizations. Using a radial grid with shifting a half mesh away from the origin, we can handle the coordinate singularity easily without pole conditions. The Sherman-Morrison formula is then applied to solve the resultant linear system in a cost-efficient way. The computational complexity of the method consists of O(MN log2 N) arithmetic operations for M × N grid points. The numerical accuracy check and some applications to the incompressible Navier-Stokes flows inside a disk are conducted.