  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.
