Superlinear convergence estimates for a conjugate gradient method for the biharmonic equation

Raymond H. CHAN; Thomas K. DELILLO; Mark A. HORN

doi:10.1137/S1064827596303570

Superlinear convergence estimates for a conjugate gradient method for the biharmonic equation

Raymond H. CHAN^*
, Thomas K. DELILLO
, Mark A. HORN

^*Corresponding author for this work

Research output: Journal Publications › Journal Article (refereed) › peer-review

5 Citations (Scopus)

Abstract

The method of Muskhelishvili for solving the biharmonic equation using conformal mapping is investigated. In [R. H. Chan, T. K. DeLillo, and M. A. Horn, SIAM J. Sci. Comput., 18 (1997), pp. 1571-1582] it was shown, using the Hankel structure, that the linear system in [N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, the Netherlands] is the discretization of the identity plus a compact operator, and therefore the conjugate gradient method will converge superlinearly. Estimates are given here of the superlinear convergence in the cases when the boundary curve is analytic or in a Hölder class.

Original language	English
Pages (from-to)	139-147
Number of pages	9
Journal	SIAM Journal on Scientific Computing
Volume	19
Issue number	1
DOIs	https://doi.org/10.1137/S1064827596303570
Publication status	Published - Jan 1998
Externally published	Yes

Keywords

Biharmonic equation
Conjugate gradient method
Hankel matrices
Numerical conformal mapping

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10.1137/S1064827596303570

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title = "Superlinear convergence estimates for a conjugate gradient method for the biharmonic equation",

abstract = "The method of Muskhelishvili for solving the biharmonic equation using conformal mapping is investigated. In [R. H. Chan, T. K. DeLillo, and M. A. Horn, SIAM J. Sci. Comput., 18 (1997), pp. 1571-1582] it was shown, using the Hankel structure, that the linear system in [N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, the Netherlands] is the discretization of the identity plus a compact operator, and therefore the conjugate gradient method will converge superlinearly. Estimates are given here of the superlinear convergence in the cases when the boundary curve is analytic or in a H{\"o}lder class.",

keywords = "Biharmonic equation, Conjugate gradient method, Hankel matrices, Numerical conformal mapping",

author = "CHAN, \{Raymond H.\} and DELILLO, \{Thomas K.\} and HORN, \{Mark A.\}",

year = "1998",

month = jan,

doi = "10.1137/S1064827596303570",

language = "English",

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T1 - Superlinear convergence estimates for a conjugate gradient method for the biharmonic equation

AU - CHAN, Raymond H.

AU - DELILLO, Thomas K.

AU - HORN, Mark A.

PY - 1998/1

Y1 - 1998/1

N2 - The method of Muskhelishvili for solving the biharmonic equation using conformal mapping is investigated. In [R. H. Chan, T. K. DeLillo, and M. A. Horn, SIAM J. Sci. Comput., 18 (1997), pp. 1571-1582] it was shown, using the Hankel structure, that the linear system in [N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, the Netherlands] is the discretization of the identity plus a compact operator, and therefore the conjugate gradient method will converge superlinearly. Estimates are given here of the superlinear convergence in the cases when the boundary curve is analytic or in a Hölder class.

AB - The method of Muskhelishvili for solving the biharmonic equation using conformal mapping is investigated. In [R. H. Chan, T. K. DeLillo, and M. A. Horn, SIAM J. Sci. Comput., 18 (1997), pp. 1571-1582] it was shown, using the Hankel structure, that the linear system in [N. I. Muskhelishvili, Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Groningen, the Netherlands] is the discretization of the identity plus a compact operator, and therefore the conjugate gradient method will converge superlinearly. Estimates are given here of the superlinear convergence in the cases when the boundary curve is analytic or in a Hölder class.

KW - Biharmonic equation

KW - Conjugate gradient method

KW - Hankel matrices

KW - Numerical conformal mapping

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U2 - 10.1137/S1064827596303570

DO - 10.1137/S1064827596303570

M3 - Journal Article (refereed)

AN - SCOPUS:2142788146

SN - 1064-8275

VL - 19

SP - 139

EP - 147

JO - SIAM Journal on Scientific Computing

JF - SIAM Journal on Scientific Computing

IS - 1

ER -

Superlinear convergence estimates for a conjugate gradient method for the biharmonic equation

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Keywords

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