Petviashvilli's Method for the Dirichlet Problem

Abstract

We examine the Petviashvilli method for solving the equation $\ddot{I}•-\hat{I}”\ddot{I}•=|\ddot{I}•{|}^{p-1}\ddot{I}•$ on a bounded domain $\hat{I}\text{c◯}\subset {\mathbb{R}}^d$ 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.

24 pages, 7 figures, shortened for publication with some corrections. See v1 for more detailed proofs of the local convergence