On the numerical solution of a semilinear elliptic eigenproblem of Lane-Emden type, (I): Problem formulation and description of the algorithms

  • In this first part of our two-part article, we present some theoretical background along with descriptions of some numerical techniques for solving a particular semilinear elliptic eigenproblem of Lane-Emden type on a triangular domain without any lines of symmetry. For solving the principal first eigenproblem, we describe an operatorsplitting method applied to the corresponding time-dependent problem. For solving higher veigenproblems, we describe an arclength continuation method applied to a particular perturbation of the original problem, which admits solution branches bifurcating from the trivial solution branch at eigenvalues of its linearization. We then solve the original eigenproblem by jumping to a point on the unperturbed solution branch from a nearby point on the corresponding continued perturbed branch, then normalizing the result. Finally, for comparison, we describe a particular implementation of Newton's method applied directly to the original constrained nonlinearIn this first part of our two-part article, we present some theoretical background along with descriptions of some numerical techniques for solving a particular semilinear elliptic eigenproblem of Lane-Emden type on a triangular domain without any lines of symmetry. For solving the principal first eigenproblem, we describe an operatorsplitting method applied to the corresponding time-dependent problem. For solving higher veigenproblems, we describe an arclength continuation method applied to a particular perturbation of the original problem, which admits solution branches bifurcating from the trivial solution branch at eigenvalues of its linearization. We then solve the original eigenproblem by jumping to a point on the unperturbed solution branch from a nearby point on the corresponding continued perturbed branch, then normalizing the result. Finally, for comparison, we describe a particular implementation of Newton's method applied directly to the original constrained nonlinear eigenproblem.show moreshow less

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Metadaten
Author:Fritz J. II. Foss, Roland GlowinskiGND, Ronald H. W. HoppeGND
URN:urn:nbn:de:bvb:384-opus4-4186
Frontdoor URLhttps://opus.bibliothek.uni-augsburg.de/opus4/517
Series (Serial Number):Preprints des Instituts für Mathematik der Universität Augsburg (2007-10)
Type:Preprint
Language:English
Publishing Institution:Universität Augsburg
Contributing Corporation:Department of Mathematics, University of Houston
Release Date:2007/05/30
Tag:semilinear eigenproblem; operator splitting; arclength continuation; least squares
GND-Keyword:Semilineares Eigenwertproblem; Operator-Splitting-Verfahren; Fortsetzungsmethode; Methode der kleinsten Quadrate
Institutes:Mathematisch-Naturwissenschaftlich-Technische Fakultät
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik
Mathematisch-Naturwissenschaftlich-Technische Fakultät / Institut für Mathematik / Lehrstuhl für Numerische Mathematik
Dewey Decimal Classification:5 Naturwissenschaften und Mathematik / 51 Mathematik / 510 Mathematik