Petviashvilli’s Method for the Dirichlet Problem

D. Olson, S. Shukla, G. Simpson, D. Spirn

Research output: Contribution to journalArticlepeer-review

8 Scopus citations


We examine Petviashvilli’s method for solving the equation ϕ-Δϕ=|ϕ|p-1ϕ on a bounded domain Ω⊂Rd with Dirichlet boundary conditions. We prove a local convergence result, using spectral analysis, akin to the result for the problem on 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.

Original languageEnglish (US)
Pages (from-to)296-320
Number of pages25
JournalJournal of Scientific Computing
Issue number1
StatePublished - Jan 1 2016

Bibliographical note

Publisher Copyright:
© 2015, Springer Science+Business Media New York.


  • Global convergence
  • Iterative methods
  • Nonlinear waves
  • Semilinear elliptic equations
  • Solitary waves


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