Symmetry and Automated Branch Following for a Semilinear Elliptic PDE on a Fractal Region

TL;DR

This paper presents a computational approach to solving a semilinear elliptic PDE on a fractal region, utilizing symmetry analysis, bifurcation theory, and automated branch following to identify multiple solution types.

Contribution

It introduces the bifurcation digraph for symmetry analysis and develops automated branch following methods for efficient solution discovery on fractal domains.

Findings

Identified 23 symmetry types of solutions.

Mapped 59 symmetry-breaking bifurcations.

Automated branch following reduces computational effort.

Abstract

We apply the Gradient-Newton-Galerkin-Algorithm (GNGA) of Neuberger & 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 post-process 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, 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.