  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.
