Petviashvilli's Method for the Dirichlet Problem

TL;DR

This paper analyzes the Petviashvilli method for solving nonlinear elliptic equations with Dirichlet boundary conditions, establishing local and global convergence results through spectral analysis and energy methods.

Contribution

It provides the first rigorous proof of both local and global convergence of the Petviashvilli method for bounded domains with Dirichlet conditions.

Findings

Proved local convergence using spectral analysis.

Established global convergence via nonlinear inequalities.

Demonstrated energy decrease along iteration sequence.

Abstract

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 a local convergence result, using spectral analysis, akin to the result for the problem on $\mathbb{R}$ by Pelinovsky & Stepanyants, 2004. 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.