A COMPACT DIFFERENCE SCHEME FOR THE BIHARMONIC EQUATION IN PLANAR IRREGULAR DOMAINS

M. Ben-Artzi; I. Chorev; J. P. Croisille; D. Fishelov

doi:10.1137/080718784

A COMPACT DIFFERENCE SCHEME FOR THE BIHARMONIC EQUATION IN PLANAR IRREGULAR DOMAINS

M. Ben-Artzi
, I. Chorev
, J. P. Croisille
, D. Fishelov

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

42 Scopus citations

Abstract

We present a finite difference scheme, applicable to general irregular planar domains, to approximate the biharmonic equation. The irregular domain is embedded in a Cartesian grid. In order to approximate Δ²Φ at a grid point we interpolate the data on the (irregular) stencil by a polynomial of degree six. The finite difference scheme is Δ²Q_Φ(0, 0), where Q_Φ is the interpolation polynomial. The interpolation polynomial is not uniquely determined. We present a method to construct such an interpolation polynomial and prove that our construction is second order accurate. For a regular stencil, [M. Ben-Artzi, J.-P. Croisille, and D. Fishelov, SIAM J. Sci. Comput., 31 (2008), pp. 303–333] shows that the proposed interpolation polynomial is fourth order accurate. We present some suitable numerical examples.

Original language	English
Pages (from-to)	3087-3108
Number of pages	22
Journal	SIAM Journal on Numerical Analysis
Volume	47
Issue number	4
DOIs	https://doi.org/10.1137/080718784
State	Published - 2009

Bibliographical note

Publisher Copyright:
© 2009 Society for Industrial and Applied Mathematics.

Keywords

biharmonic problem
compact approximations
finite differences
high accuracy
irregular domain

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10.1137/080718784

Cite this

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abstract = "We present a finite difference scheme, applicable to general irregular planar domains, to approximate the biharmonic equation. The irregular domain is embedded in a Cartesian grid. In order to approximate Δ2Φ at a grid point we interpolate the data on the (irregular) stencil by a polynomial of degree six. The finite difference scheme is Δ2QΦ(0, 0), where QΦ is the interpolation polynomial. The interpolation polynomial is not uniquely determined. We present a method to construct such an interpolation polynomial and prove that our construction is second order accurate. For a regular stencil, [M. Ben-Artzi, J.-P. Croisille, and D. Fishelov, SIAM J. Sci. Comput., 31 (2008), pp. 303–333] shows that the proposed interpolation polynomial is fourth order accurate. We present some suitable numerical examples.",

keywords = "biharmonic problem, compact approximations, finite differences, high accuracy, irregular domain",

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AU - Fishelov, D.

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AB - We present a finite difference scheme, applicable to general irregular planar domains, to approximate the biharmonic equation. The irregular domain is embedded in a Cartesian grid. In order to approximate Δ2Φ at a grid point we interpolate the data on the (irregular) stencil by a polynomial of degree six. The finite difference scheme is Δ2QΦ(0, 0), where QΦ is the interpolation polynomial. The interpolation polynomial is not uniquely determined. We present a method to construct such an interpolation polynomial and prove that our construction is second order accurate. For a regular stencil, [M. Ben-Artzi, J.-P. Croisille, and D. Fishelov, SIAM J. Sci. Comput., 31 (2008), pp. 303–333] shows that the proposed interpolation polynomial is fourth order accurate. We present some suitable numerical examples.

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KW - finite differences

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A COMPACT DIFFERENCE SCHEME FOR THE BIHARMONIC EQUATION IN PLANAR IRREGULAR DOMAINS

Abstract

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Keywords

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