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Computing 9 by Springer-Verlag 1978 Numerical Solution of Nonlinear Elliptic Partial Differential Equations by a Generalized Conjugate Gradient Method (1977)
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@MISC{77computing9,
author = {},
title = {Computing 9 by Springer-Verlag 1978 Numerical Solution of Nonlinear Elliptic Partial Differential Equations by a Generalized Conjugate Gradient Method},
year = {1977}
}
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Abstract
Abstract-- Zusammenfassung Numerical Solution of Nonlinear Elliptic Partial Differential Equations by a Generalized Conjugate Gradient Method. We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partialdifferential boundary-value problems. Here, extensions to the nonlinear case are considered. We split the original discretized operator into the sum of two operators, one of which corresponds to a more easily solvable system of equations, and accelerate the associated iteration based on this splitting by (nonlinear) conjugate gradients. The behavior of the method is illustrated for the minimal surface equation with splittings corresponding to nonlinear SSOR, to approximate factorization of the Jacobian matrix, and to elliptic operators suitable for use with fast direct methods. The results of numerical experiments are given as well for a mildy nonlinear example, for which, in the corresponding linear case, the finite termination property of the conjugate gradient algorithm is crucial. Numerische L6sung von nichtlinearen elliptischen partiellen Differentialgleichungen mit einer verallgemeinerten Methode der konjugierten Gradienten. Wir haben friiher eine verallgemeinerte Methode der konjugierten Gradienten studiert, um diinnbesetzte positiv definite Systeme von linearen Gleichungen
Keyphrases
generalized conjugate gradient method nonlinear elliptic partial differential equation numerical solution solvable system numerical experiment conjugate gradient algorithm abstract zusammenfassung numerical solution numerische l6sung von linear equation elliptic operator conjugate gradient elliptic partialdifferential boundary-value problem fast direct method original discretized operator jacobian matrix minimal surface equation sparse positive-definite system mildy nonlinear example corresponding linear case nonlinear case finite termination property
