On the numerical solution of the biharmonic equation in the plane

Anne Greenbaum, Leslie Greengard, Anita Mayo

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)216-225
Number of pages10
JournalPhysica D: Nonlinear Phenomena
Volume60
Issue number1-4
DOIs
StatePublished - Nov 1 1992

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biharmonic equations
Biharmonic Equation
Integral equations
integral equations
Integral Equations
Numerical Solution
Plane Elasticity
Fredholm Equation
Integral Equation Method
Incompressible Viscous Flow
Moving Boundary
Conjugate Gradient
Fredholm equations
Applied mathematics
Free Boundary
Integral Operator
Fast Algorithm
Convergence Rate
incompressible flow
free boundaries

ASJC Scopus subject areas

  • Applied Mathematics
  • Statistical and Nonlinear Physics

Cite this

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

In: Physica D: Nonlinear Phenomena, Vol. 60, No. 1-4, 01.11.1992, p. 216-225.

Research output: Contribution to journalArticle

Greenbaum, Anne ; Greengard, Leslie ; Mayo, Anita. / On the numerical solution of the biharmonic equation in the plane. In: Physica D: Nonlinear Phenomena. 1992 ; Vol. 60, No. 1-4. pp. 216-225.
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