### Abstract

We apply the gradient Newton-Galerkin algorithm (GNGA) of Neuberger and Swift to find solutions to a semilinear elliptic Dirichlet problem on the region whose boundary is the Koch snowflake. In a recent paper, we described an accurate and efficient method for generating a basis of eigenfunctions of the Laplacian on this region. In that work, we used the symmetry of the snowflake region to analyze and postprocess the basis, rendering it suitable for input to the GNGA, The GNGA uses Newton's method on the eigenfunction expansion coefficients to find solutions to the semilinear problem. This article introduces the bifurcation digraph, an extension of the lattice of isotropy subgroups. For our example, the bifurcation digraph shows the 23 possible symmetry types of solutions to the PDE, and the 59 generic symmetry-breaking bifurcations among these symmetry types. Our numerical code uses continuation methods and follows branches created at symmetry-breaking bifurcations, and so the human user does not need to supply initial guesses for Newton's method. Starting from the known trivial solution, the code automatically finds at least one solution with each of the symmetry types that we predict can exist. Such computationally intensive investigations necessitated the writing of automated branch following code, whereby symmetry information was used to reduce the number of computations per GNGA execution and to make intelligent branch following decisions at bifurcation points.

Original language | English (US) |
---|---|

Pages (from-to) | 476-507 |

Number of pages | 32 |

Journal | SIAM Journal on Applied Dynamical Systems |

Volume | 5 |

Issue number | 3 |

DOIs | |

State | Published - 2006 |

### Fingerprint

### Keywords

- Bifurcation
- GNGA
- Semilinear elliptic pde
- Snowflake
- Symmetry

### ASJC Scopus subject areas

- Analysis
- Modeling and Simulation

### Cite this

**Symmetry and automated branch following for a Semilinear elliptic PDE on a fractal region.** / Neuberger, John M; Sieben, Nandor; Swift, James W.

Research output: Contribution to journal › Article

}

TY - JOUR

T1 - Symmetry and automated branch following for a Semilinear elliptic PDE on a fractal region

AU - Neuberger, John M

AU - Sieben, Nandor

AU - Swift, James W

PY - 2006

Y1 - 2006

N2 - We apply the gradient Newton-Galerkin algorithm (GNGA) of Neuberger and Swift to find solutions to a semilinear elliptic Dirichlet problem on the region whose boundary is the Koch snowflake. In a recent paper, we described an accurate and efficient method for generating a basis of eigenfunctions of the Laplacian on this region. In that work, we used the symmetry of the snowflake region to analyze and postprocess the basis, rendering it suitable for input to the GNGA, The GNGA uses Newton's method on the eigenfunction expansion coefficients to find solutions to the semilinear problem. This article introduces the bifurcation digraph, an extension of the lattice of isotropy subgroups. For our example, the bifurcation digraph shows the 23 possible symmetry types of solutions to the PDE, and the 59 generic symmetry-breaking bifurcations among these symmetry types. Our numerical code uses continuation methods and follows branches created at symmetry-breaking bifurcations, and so the human user does not need to supply initial guesses for Newton's method. Starting from the known trivial solution, the code automatically finds at least one solution with each of the symmetry types that we predict can exist. Such computationally intensive investigations necessitated the writing of automated branch following code, whereby symmetry information was used to reduce the number of computations per GNGA execution and to make intelligent branch following decisions at bifurcation points.

AB - We apply the gradient Newton-Galerkin algorithm (GNGA) of Neuberger and Swift to find solutions to a semilinear elliptic Dirichlet problem on the region whose boundary is the Koch snowflake. In a recent paper, we described an accurate and efficient method for generating a basis of eigenfunctions of the Laplacian on this region. In that work, we used the symmetry of the snowflake region to analyze and postprocess the basis, rendering it suitable for input to the GNGA, The GNGA uses Newton's method on the eigenfunction expansion coefficients to find solutions to the semilinear problem. This article introduces the bifurcation digraph, an extension of the lattice of isotropy subgroups. For our example, the bifurcation digraph shows the 23 possible symmetry types of solutions to the PDE, and the 59 generic symmetry-breaking bifurcations among these symmetry types. Our numerical code uses continuation methods and follows branches created at symmetry-breaking bifurcations, and so the human user does not need to supply initial guesses for Newton's method. Starting from the known trivial solution, the code automatically finds at least one solution with each of the symmetry types that we predict can exist. Such computationally intensive investigations necessitated the writing of automated branch following code, whereby symmetry information was used to reduce the number of computations per GNGA execution and to make intelligent branch following decisions at bifurcation points.

KW - Bifurcation

KW - GNGA

KW - Semilinear elliptic pde

KW - Snowflake

KW - Symmetry

UR - http://www.scopus.com/inward/record.url?scp=57249111985&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=57249111985&partnerID=8YFLogxK

U2 - 10.1137/050640485

DO - 10.1137/050640485

M3 - Article

VL - 5

SP - 476

EP - 507

JO - SIAM Journal on Applied Dynamical Systems

JF - SIAM Journal on Applied Dynamical Systems

SN - 1536-0040

IS - 3

ER -