Petviashvilli's Method for the Dirichlet Problem
Provides rigorous convergence guarantees for a numerical method used in nonlinear PDEs, extending prior results from the real line to bounded domains.
The authors prove local and global convergence of Petviashvilli's method for a semilinear elliptic PDE with Dirichlet boundary conditions, establishing monotonic decay of an energy functional along the iteration sequence.
We examine the Petviashvilli method for solving the equation $ ϕ- Δϕ= |ϕ|^{p-1} ϕ$ on a bounded domain $Ω\subset \mathbb{R}^d$ 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.