Petviashvilli's Method for the Dirichlet Problem

Derek Olson, Soumitra Shukla, Gideon Simpson, Daniel Spirn

Published 2014-11-15Version 1

We examine the Petviashvilli method for solving the equation $ \phi - \Delta \phi = |\phi|^{p-1} \phi$ on a bounded domain $\Omega \subset \mathbb{R}^d$ with Dirichlet boundary conditions. We prove that a ground state is a local attractor of the iteration procedure, similar to the results on $\mathbb{R}^d$ by Pelinovsky and Stepanyants (2004). We prove a global convergence result by generating a suite of nonlinear inequalities for the iteration sequence, and we show that the sequence has a natural energy that decreases along the sequence.