Abstract
We have studied previously a generalized conjugate gradient method for solving sparse positive-definite systems of linear equations arising from the discretization of elliptic partial-differential 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.
Zusammenfassung
Wir haben früher eine verallgemeinerte Methode der konjugierten Gradienten studiert, um dünnbesetzte positiv definite Systeme von linearen Gleichungen zu lösen, die von der Diskretisierung von elliptischen partiellen Differential-Randwertproblemen herrühren. Wir betrachten hier die Verallgemeinerung auf den nichtlinearen Fall: Wir spalten den ursprünglichen diskretisierten Operator auf in eine Summe von zwei Operatoren. Einer von diesen Operatoren entspricht einem leicht lösbaren System von Gleichungen, und wir beschleunigen die aus dieser Spaltung hervorgehende Iteration mit (nichtlinearen) konjugierten Gradienten. Das Verhalten der Methode wird illustriert durch Anwendung auf die Minimalflächen-Gleichung, mit Spaltungen entsprechend dem nichtlinearen SSOR-Verfahren, der angenäherten Faktorisierung der Jacobi-Matrix, oder den elliptischen Operatoren, die sich für schnelle direkte Methoden eignen. Die Resultate von numerischen Experimenten für ein nur schwach nichtlineares Beispiel sind ebenfalls angegeben. Für den entsprechenden linearen Fall ist in diesem Fall die Konvergenz des konjugierten Gradienten-Algorithmus in einer endlichen Anzahl von Schritten wesentlich.
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Concus, P., Golub, G.H. & O'Leary, D.P. Numerical solution of nonlinear elliptic partial differential equations by a generalized conjugate gradient method. Computing 19, 321–339 (1978). https://doi.org/10.1007/BF02252030
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DOI: https://doi.org/10.1007/BF02252030