Newton's method and symmetry for semilinear elliptic pde on the cube

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Abstract

We seek discrete approximations to solutions u : Ω → R of semilinear elliptic PDE of the form δu + fs(u) = 0, where fs is a one-parameter family of nonlinear functions and Ω is a domain in Rd. The main achievement of this paper is the approximation of solutions to the PDE on the cube Ω = (0, π)3 -R3. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends our work in [Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), pp. 2531-2556], wherein we combined symmetry analysis with modified implementations of the gradient Newton-Galerkin algorithm (GNGA [J. M. Neuberger and J. W. Swift, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801-820]) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our previous paper [Int. J. Parallel Program., 40 (2012), pp. 443-464].

Original languageEnglish (US)
Pages (from-to)1237-1279
Number of pages43
JournalSIAM Journal on Applied Dynamical Systems
Volume12
Issue number3
DOIs
StatePublished - 2013

Fingerprint

Elliptic PDE
Newton-Raphson method
Semilinear
Newton Methods
Chaos theory
Regular hexahedron
Bifurcation
Eigenspace
Isotropy
Symmetry
Equivalence classes
Bifurcation (mathematics)
Chaos
Subgroup
Discrete Approximation
Parallel Programs
Bifurcation Point
Bifurcation Diagram
Conjugacy class
Symmetry Group

Keywords

  • Bifurcation with symmetry
  • Cube
  • GNGA
  • Numerical methods
  • Semilinear elliptic PDE

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation

Cite this

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title = "Newton's method and symmetry for semilinear elliptic pde on the cube",
abstract = "We seek discrete approximations to solutions u : Ω → R of semilinear elliptic PDE of the form δu + fs(u) = 0, where fs is a one-parameter family of nonlinear functions and Ω is a domain in Rd. The main achievement of this paper is the approximation of solutions to the PDE on the cube Ω = (0, π)3 -R3. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends our work in [Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), pp. 2531-2556], wherein we combined symmetry analysis with modified implementations of the gradient Newton-Galerkin algorithm (GNGA [J. M. Neuberger and J. W. Swift, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801-820]) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our previous paper [Int. J. Parallel Program., 40 (2012), pp. 443-464].",
keywords = "Bifurcation with symmetry, Cube, GNGA, Numerical methods, Semilinear elliptic PDE",
author = "Neuberger, {John M} and Nandor Sieben and Swift, {James W}",
year = "2013",
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language = "English (US)",
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T1 - Newton's method and symmetry for semilinear elliptic pde on the cube

AU - Neuberger, John M

AU - Sieben, Nandor

AU - Swift, James W

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N2 - We seek discrete approximations to solutions u : Ω → R of semilinear elliptic PDE of the form δu + fs(u) = 0, where fs is a one-parameter family of nonlinear functions and Ω is a domain in Rd. The main achievement of this paper is the approximation of solutions to the PDE on the cube Ω = (0, π)3 -R3. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends our work in [Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), pp. 2531-2556], wherein we combined symmetry analysis with modified implementations of the gradient Newton-Galerkin algorithm (GNGA [J. M. Neuberger and J. W. Swift, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801-820]) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our previous paper [Int. J. Parallel Program., 40 (2012), pp. 443-464].

AB - We seek discrete approximations to solutions u : Ω → R of semilinear elliptic PDE of the form δu + fs(u) = 0, where fs is a one-parameter family of nonlinear functions and Ω is a domain in Rd. The main achievement of this paper is the approximation of solutions to the PDE on the cube Ω = (0, π)3 -R3. There are 323 possible isotropy subgroups of functions on the cube, which fall into 99 conjugacy classes. The bifurcations with symmetry in this problem are quite interesting, including many with 3-dimensional critical eigenspaces. Our automated symmetry analysis is necessary with so many isotropy subgroups and bifurcations among them, and it allows our code to follow one branch in each equivalence class that is created at a bifurcation point. Our most complicated result is the complete analysis of a degenerate bifurcation with a 6-dimensional critical eigenspace. This article extends our work in [Internat. J. Bifur. Chaos Appl. Sci. Engrg., 19 (2009), pp. 2531-2556], wherein we combined symmetry analysis with modified implementations of the gradient Newton-Galerkin algorithm (GNGA [J. M. Neuberger and J. W. Swift, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 11 (2001), pp. 801-820]) to automatically generate bifurcation diagrams and solution graphics for small, discrete problems with large symmetry groups. The code described in the current paper is efficiently implemented in parallel, allowing us to investigate a relatively fine-mesh discretization of the cube. We use the methodology and corresponding library presented in our previous paper [Int. J. Parallel Program., 40 (2012), pp. 443-464].

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