### 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 language | English (US) |
---|---|

Pages (from-to) | 1237-1279 |

Number of pages | 43 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 12 |

Issue number | 3 |

DOIs | |

State | Published - 2013 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Analysis
- Modeling and Simulation

### Cite this

**Newton's method and symmetry for semilinear elliptic pde on the cube.** / Neuberger, John M; Sieben, Nandor; Swift, James W.

Research output: Contribution to journal › Article

*SIAM Journal on Applied Dynamical Systems*, vol. 12, no. 3, pp. 1237-1279. https://doi.org/10.1137/120899054

}

TY - JOUR

T1 - Newton's method and symmetry for semilinear elliptic pde on the cube

AU - Neuberger, John M

AU - Sieben, Nandor

AU - Swift, James W

PY - 2013

Y1 - 2013

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].

KW - Bifurcation with symmetry

KW - Cube

KW - GNGA

KW - Numerical methods

KW - Semilinear elliptic PDE

UR - http://www.scopus.com/inward/record.url?scp=84884971265&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84884971265&partnerID=8YFLogxK

U2 - 10.1137/120899054

DO - 10.1137/120899054

M3 - Article

AN - SCOPUS:84884971265

VL - 12

SP - 1237

EP - 1279

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 3

ER -