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.…
| Author: | Fritz J. II. Foss, Roland GlowinskiGND, Ronald H. W. HoppeORCiDGND |
|---|---|
| URN: | urn:nbn:de:bvb:384-opus4-4186 |
| Frontdoor URL | https://opus.bibliothek.uni-augsburg.de/opus4/517 |
| Series (Serial Number): | Preprints des Instituts für Mathematik der Universität Augsburg (2007-10) |
| Type: | Preprint |
| Language: | English |
| Date of Publication (online): | 2007/05/30 |
| Publishing Institution: | Universität Augsburg |
| Contributing Corporation: | Department of Mathematics, University of Houston |
| Release Date: | 2007/05/30 |
| Tag: | arclength continuation; least squares; operator splitting; semilinear eigenproblem |
| 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 |
