## Abstract

The solution to the biharmonic equation in a simply connected region **Ω** in the plane is computed in terms of the Goursat functions. The boundary conditions are conformally transplanted to the disk with a numerical conformal map. A linear system is obtained for the Taylor coefficients of the Goursat functions. The coefficient matrix of the linear system can be put in the form * I *+

*, where K is the discretization of a compact operator. K can be thought of as the composition of a block Hankel matrix with a diagonal matrix. The compactness leads to clustering of eigenvalues, and the Hankel structure yields a matrix-vector multiplication cost of*

**K****O(**. Thus, if the conjugate gradient method is applied to the system, then superlinear convergence will be obtained. Numerical results are given to illustrate the spectrum clustering and superlinear convergence.

*N*log*N*)Original language | English |
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Pages (from-to) | 1571-1582 |

Number of pages | 12 |

Journal | SIAM Journal on Scientific Computing |

Volume | 18 |

Issue number | 6 |

DOIs | |

Publication status | Published - Nov 1997 |

Externally published | Yes |

## Keywords

- Biharmonic equation
- Hankel matrices
- Numerical conformal mapping