### Abstract

The biharmonic equation arises in a variety of problems in applied mathematics, most notably in plane elasticity and in viscous incompressible flow. Integral equation methods are natural candidates for the numerical solution of such problems, since they discritize the boundary alone, are easy to apply in the case of free or moving boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains, regardless of shape. In this paper, we follow the work of Mayo and Greenbaum and make use of the Sherman-Lauricella integral equation which is a Fredholm equation with bounded kernel. We describe a fast algorithm for the evaluation of the integral operators appearing in that equation. When combined with a conjugate gradient like algorithm, we are able to solve the discretized integral equation in an amount of time proportional to N, where N is the number of nodes in the discretization of the boundary.

Original language | English (US) |
---|---|

Pages (from-to) | 216-225 |

Number of pages | 10 |

Journal | Physica D: Nonlinear Phenomena |

Volume | 60 |

Issue number | 1-4 |

DOIs | |

State | Published - Nov 1 1992 |

### Fingerprint

### ASJC Scopus subject areas

- Applied Mathematics
- Statistical and Nonlinear Physics

### Cite this

*Physica D: Nonlinear Phenomena*,

*60*(1-4), 216-225. https://doi.org/10.1016/0167-2789(92)90238-I

**On the numerical solution of the biharmonic equation in the plane.** / Greenbaum, Anne; Greengard, Leslie; Mayo, Anita.

Research output: Contribution to journal › Article

*Physica D: Nonlinear Phenomena*, vol. 60, no. 1-4, pp. 216-225. https://doi.org/10.1016/0167-2789(92)90238-I

}

TY - JOUR

T1 - On the numerical solution of the biharmonic equation in the plane

AU - Greenbaum, Anne

AU - Greengard, Leslie

AU - Mayo, Anita

PY - 1992/11/1

Y1 - 1992/11/1

N2 - The biharmonic equation arises in a variety of problems in applied mathematics, most notably in plane elasticity and in viscous incompressible flow. Integral equation methods are natural candidates for the numerical solution of such problems, since they discritize the boundary alone, are easy to apply in the case of free or moving boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains, regardless of shape. In this paper, we follow the work of Mayo and Greenbaum and make use of the Sherman-Lauricella integral equation which is a Fredholm equation with bounded kernel. We describe a fast algorithm for the evaluation of the integral operators appearing in that equation. When combined with a conjugate gradient like algorithm, we are able to solve the discretized integral equation in an amount of time proportional to N, where N is the number of nodes in the discretization of the boundary.

AB - The biharmonic equation arises in a variety of problems in applied mathematics, most notably in plane elasticity and in viscous incompressible flow. Integral equation methods are natural candidates for the numerical solution of such problems, since they discritize the boundary alone, are easy to apply in the case of free or moving boundaries, and achieve superalgebraic convergence rates on sufficiently smooth domains, regardless of shape. In this paper, we follow the work of Mayo and Greenbaum and make use of the Sherman-Lauricella integral equation which is a Fredholm equation with bounded kernel. We describe a fast algorithm for the evaluation of the integral operators appearing in that equation. When combined with a conjugate gradient like algorithm, we are able to solve the discretized integral equation in an amount of time proportional to N, where N is the number of nodes in the discretization of the boundary.

UR - http://www.scopus.com/inward/record.url?scp=0001255195&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0001255195&partnerID=8YFLogxK

U2 - 10.1016/0167-2789(92)90238-I

DO - 10.1016/0167-2789(92)90238-I

M3 - Article

VL - 60

SP - 216

EP - 225

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-4

ER -