### 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 log_{2} 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 language | English |
---|---|

Pages (from-to) | 679-695 |

Number of pages | 17 |

Journal | Applied Mathematics and Computation |

Volume | 164 |

Issue number | 3 |

DOIs | |

State | Published - 25 May 2005 |

### Keywords

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