Numerical approximation of the first eigenpair of the p-Laplacian using finite elements and the penalty method

Lew Lefton, Dongming Wei

Research output: Contribution to journalArticle

31 Citations (Scopus)

Abstract

For 1 <p <∞, consider the eigenvalue problem for the p-Laplacian - △pu = λ|u|p-2u, u|∂Ω = 0 where △pu = div(|▽u|p-2▽u). The first eigenvalue λ1 can be obtained by minimizing the functional ∫Ω|▽u|p/ ∫Ω |u|p over W1,P 0(Ω). A method for computing λ1 numerically is presented. The technique uses a finite element approximation to the first eigenfunction and a penalty function to enforce the constraint. Convergence is proved and numerical results are presented. The numerical results are compared with exact values when known. A lower bound for p-Laplacian eigenvalues is also presented. In particular, this work provides a computational framework for obtaining precise approximations of the best constant for the Sobolev imbedding W1,p 0(Ω) (Rightwards arrow with hook sign) Lp(Ω).

Original languageEnglish
Pages (from-to)389-399
Number of pages11
JournalNumerical Functional Analysis and Optimization
Volume18
Issue number3-4
Publication statusPublished - 1997
Externally publishedYes

Fingerprint

Hooks
Penalty Method
P-Laplacian
Numerical Approximation
Eigenvalues and eigenfunctions
Finite Element
Laplacian Eigenvalues
Numerical Results
Best Constants
Imbedding
First Eigenvalue
Penalty Function
Finite Element Approximation
Eigenvalue Problem
Eigenfunctions
Lower bound
Computing
Approximation
Framework

Keywords

  • Finite elements
  • Nonlinear eigenvalue problems
  • p-Laplacian
  • Penalty method

ASJC Scopus subject areas

  • Applied Mathematics
  • Control and Optimization

Cite this

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AB - For 1 pu = λ|u|p-2u, u|∂Ω = 0 where △pu = div(|▽u|p-2▽u). The first eigenvalue λ1 can be obtained by minimizing the functional ∫Ω|▽u|p/ ∫Ω |u|p over W1,P 0(Ω). A method for computing λ1 numerically is presented. The technique uses a finite element approximation to the first eigenfunction and a penalty function to enforce the constraint. Convergence is proved and numerical results are presented. The numerical results are compared with exact values when known. A lower bound for p-Laplacian eigenvalues is also presented. In particular, this work provides a computational framework for obtaining precise approximations of the best constant for the Sobolev imbedding W1,p 0(Ω) (Rightwards arrow with hook sign) Lp(Ω).

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