Newton's Method and Symmetry for Semilinear Elliptic PDE on the Cube

作者：Neuberger John M^*; Sieben Nandor; Swift James W

来源：SIAM Journal on Applied Dynamical Systems, 2013, 12(3): 1237-1279.

DOI：10.1137/120899054

• bifurcation with symmetry
• semilinear elliptic PDE
• cube
• GNGA
• numerical methods

We seek discrete approximations to solutions u : Omega -> R of semilinear elliptic PDE of the form Delta u + f(s)(u) = 0, where f(s) is a one-parameter family of nonlinear functions and Omega is a domain in R-d. The main achievement of this paper is the approximation of solutions to the PDE on the cube Omega = (0, pi)(3) subset of R-3. 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.

• 出版日期2013