GNGA for general regions: Semilinear elliptic PDE and crossing eigenvalues

Jay L. Hineman, John M Neuberger

Research output: Contribution to journalArticle

2 Scopus citations

Abstract

We consider the semilinear elliptic PDE Δu + f(λ, u) = 0 with the zero-Dirichlet boundary condition on a family of regions, namely stadions. Linear problems on such regions have been widely studied in the past. We seek to observe the corresponding phenomena in our nonlinear setting. Using the Gradient Newton Galerkin Algorithm (GNGA) of Neuberger and Swift, we document bifurcation, nodal structure, and symmetry of solutions. This paper provides the first published instance where the GNGA is applied to general regions. Our investigation involves both the dimension of the stadions and the value λ as parameters. We find that the so-called crossings and avoided crossings of eigenvalues as the dimension of the stadions vary influences the symmetry and variational structure of nonlinear solutions in a natural way.

Original languageEnglish (US)
Pages (from-to)447-464
Number of pages18
JournalCommunications in Nonlinear Science and Numerical Simulation
Volume12
Issue number4
DOIs
StatePublished - Jul 2007

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Keywords

  • Bifurcation
  • GNGA
  • Newton's method
  • Semilinear elliptic BVP
  • Stadia
  • Stadium
  • Variational method

ASJC Scopus subject areas

  • Mechanical Engineering
  • Statistical and Nonlinear Physics

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