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On the Increase of Convergence Rates of Relaxation Procedures for Elliptic Partial Difference Equations

Authors:

M. L. Juncosa,

T. W. MullikinAuthors Info & Claims

Journal of the ACM (JACM), Volume 7, Issue 1

Pages 29 - 36

https://doi.org/10.1145/321008.321012

Published: 01 January 1960 Publication History

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Abstract

Occasionally in the numerical solution of elliptic partial differential equations the rate of convergence of relaxation methods to the solution is adversely affected by the relative proximity of certain points in the grid. It has been proposed that the removal of the unknown functional values at these points by Gaussian elimination may accelerate the convergence.

By application of the Perron-Frobenius theory of non-negative matrices it is shown that the rates of convergence of the Jacobi-Richardson and Gauss-Seidel iterations are not decreased and could be increased by this elimination. Although this may indicate that the elimination could improve the convergence rate for overrelaxation, it is still strictly an unsolved problem.

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Lin HLei ZDing MWang HShi T(2019)Fast Computation of Electrostatic Interactions for a Charged Polymer with Applied FieldChinese Journal of Polymer Science10.1007/s10118-020-2343-8Online publication date: 11-Oct-2019
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Adsuara JCordero-Carrión ICerdá-Durán PAloy M(2016)Scheduled Relaxation Jacobi methodJournal of Computational Physics10.1016/j.jcp.2016.05.053321:C(369-413)Online publication date: 15-Sep-2016
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Liu Q(2009)On the improved Gauss-Seidel method for linear systemsProceedings of the 3rd WSEAS international conference on Circuits, systems, signal and telecommunications10.5555/1519489.1519509(105-109)Online publication date: 10-Jan-2009
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Index Terms

On the Increase of Convergence Rates of Relaxation Procedures for Elliptic Partial Difference Equations
1. Mathematics of computing
  1. Mathematical analysis
    1. Differential equations
      1. Ordinary differential equations
      2. Partial differential equations

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Published In

Journal of the ACM Volume 7, Issue 1

Jan. 1960

79 pages

ISSN:0004-5411

EISSN:1557-735X

DOI:10.1145/321008

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 01 January 1960

Published in JACM Volume 7, Issue 1

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Lin HLei ZDing MWang HShi T(2019)Fast Computation of Electrostatic Interactions for a Charged Polymer with Applied FieldChinese Journal of Polymer Science10.1007/s10118-020-2343-8Online publication date: 11-Oct-2019
https://doi.org/10.1007/s10118-020-2343-8
Adsuara JCordero-Carrión ICerdá-Durán PAloy M(2016)Scheduled Relaxation Jacobi methodJournal of Computational Physics10.1016/j.jcp.2016.05.053321:C(369-413)Online publication date: 15-Sep-2016
https://dl.acm.org/doi/10.1016/j.jcp.2016.05.053
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https://dl.acm.org/doi/10.5555/1519489.1519509
Liu Q(2008)Some preconditioning techniques for linear systemsWSEAS Transactions on Mathematics10.5555/1503527.15035317:9(579-588)Online publication date: 1-Sep-2008
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Noutsos DTzoumas M(2006)On optimal improvements of classical iterative schemes for Z-matricesJournal of Computational and Applied Mathematics10.5555/1140421.1716928188:1(89-106)Online publication date: 1-Apr-2006
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Alanelli MHadjidimos A(2004)Block Gauss elimination followed by a classical iterative method for the solution of linear systemsJournal of Computational and Applied Mathematics10.1016/j.cam.2003.08.045163:2(381-400)Online publication date: 15-Feb-2004
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Hadjidimos ANoutsos DTzoumas M(2003)More on modifications and improvements of classical iterative schemes for M-matricesLinear Algebra and its Applications10.1016/S0024-3795(02)00570-0364(253-279)Online publication date: May-2003
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Milaszewicz J(1987)Improving Jacobi and Gauss-Seidel IterationsLinear Algebra and its Applications10.1016/S0024-3795(87)90321-193(161-170)Online publication date: Aug-1987
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Abstract

References

Cited By

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Recommendations

Semismooth Karush-Kuhn-Tucker Equations and Convergence Analysis of Newton and Quasi-Newton Methods for Solving these Equations

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