Petviashvilli’s Method for the Dirichlet Problem

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

doi:10.1007/s10915-015-0023-6

Petviashvilli’s Method for the Dirichlet Problem

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

Mathematics

Research output: Contribution to journal › Article › peer-review

6 Scopus citations

Abstract

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 language	English (US)
Pages (from-to)	296-320
Number of pages	25
Journal	Journal of Scientific Computing
Volume	66
Issue number	1
DOIs	https://doi.org/10.1007/s10915-015-0023-6
State	Published - Jan 1 2016

Bibliographical note

Funding Information:
The authors are grateful for several helpful conversations with Svitlana Mayboroda. D.Olson was supported by the Department of Defense (DoD) through the National Defense Science and Engineering Graduate Fellowship (NDSEG) Program. S.Shukla was supported by University of Minnesota UROP-11133. G.Simpson began this work under the support of the DOE DE-SC0002085 and the NSF PIRE OISE-0967140, and completed it under NSF DMS-1409018. D.Spirn was supported by NSF DMS-0955687.

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

Keywords

Global convergence
Iterative methods
Nonlinear waves
Semilinear elliptic equations
Solitary waves

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abstract = "We examine Petviashvilli{\textquoteright}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.",

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author = "D. Olson and S. Shukla and G. Simpson and D. Spirn",

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AU - Simpson, G.

AU - Spirn, D.

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AB - 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.

KW - Global convergence

KW - Iterative methods

KW - Nonlinear waves

KW - Semilinear elliptic equations

KW - Solitary waves

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Petviashvilli’s Method for the Dirichlet Problem

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