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

Ming-Chih Lai*, Hsi Chi Liu

*Corresponding author for this work

Research output: Contribution to journalArticle

26 Scopus citations

Abstract

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.

Original languageEnglish
Pages (from-to)679-695
Number of pages17
JournalApplied Mathematics and Computation
Volume164
Issue number3
DOIs
StatePublished - 25 May 2005

Keywords

  • Biharmonic equation
  • FFT
  • Polar coordinates
  • Sherman-Morrison formula
  • Vorticity stream function formulation

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