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

School of Mathematics

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

9 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

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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10.1007/s10915-015-0023-6

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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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AU - Olson, D.

AU - Shukla, S.

AU - Simpson, G.

AU - Spirn, D.

PY - 2016/1/1

Y1 - 2016/1/1

N2 - 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.

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

Abstract

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