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

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4 Citations (Scopus)

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 languageEnglish (US)
Pages (from-to)476-507
Number of pages32
JournalSIAM Journal on Applied Dynamical Systems
Volume5
Issue number3
DOIs
StatePublished - 2006

Fingerprint

Elliptic PDE
Semilinear
Fractals
Fractal
Branch
Galerkin
Symmetry
Bifurcation
Gradient
Newton-Raphson method
Eigenvalues and eigenfunctions
Newton Methods
Symmetry Breaking
Digraph
Continuation Method
Eigenfunction Expansion
Isotropy
Bifurcation Point
Guess
Elliptic Problems

Keywords

  • Bifurcation
  • GNGA
  • Semilinear elliptic pde
  • Snowflake
  • Symmetry

ASJC Scopus subject areas

  • Analysis
  • Modeling and Simulation

Cite this

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title = "Symmetry and automated branch following for a Semilinear elliptic PDE on a fractal region",
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.",
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author = "Neuberger, {John M} and Nandor Sieben and Swift, {James W}",
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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.

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