Petviashvilli’s Method for the Dirichlet Problem

Abstract

We examine Petviashvilli's method for solving the equation $$ \phi - \Delta \phi = |\phi |^{p-1} \phi $$?-Δ?=|?|p-1? on a bounded domain $$\Omega \subset \mathbb {R}^d$$Ω?Rd with Dirichlet boundary conditions. We prove a local convergence result, using spectral analysis, akin to the result for the problem on $$\mathbb {R}$$R by Pelinovsky and Stepanyants in [16]. We also 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.