Newton's method and Morse index for semilinear elliptic PDES

Research output: Contribution to journalArticle

21 Citations (Scopus)

Abstract

In this paper we primarily consider the family of elliptic PDEs Δu + f(u) = 0 on the square region Ω = (0, 1) × (0, 1) with zero Dirichlet boundary condition. Following our previous analysis and numerical approximations which relied on the variational characterization of solutions as critical points of an "action" functional, we consider Newton's method on the gradient of that functional. We use a Galerkin expansion, in eigenfunctions of the Laplacian, to find solutions of arbitrary Morse index. Taking f′(0) to be a bifurcation parameter, we analyze the bifurcations from the trivial solution, u ≡ 0, using symmetry arguments and our numerical algorithm. The Morse index of the approximated solutions is provided and support is found concerning several existence and nodal structure conjectures. We discuss the applicability of this method to find critical points of functional in general.

Original languageEnglish (US)
Pages (from-to)801-820
Number of pages20
JournalInternational Journal of Bifurcation and Chaos
Volume11
Issue number3
DOIs
StatePublished - Mar 2001

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Morse Index
Newton-Raphson method
Semilinear
Newton Methods
Critical point
Bifurcation
Elliptic PDE
Numerical Approximation
Galerkin
Numerical Algorithms
Dirichlet Boundary Conditions
Eigenfunctions
Trivial
Gradient
Symmetry
Eigenvalues and eigenfunctions
Zero
Arbitrary
Boundary conditions
Family

ASJC Scopus subject areas

  • General
  • Applied Mathematics

Cite this

Newton's method and Morse index for semilinear elliptic PDES. / Neuberger, John M; Swift, James W.

In: International Journal of Bifurcation and Chaos, Vol. 11, No. 3, 03.2001, p. 801-820.

Research output: Contribution to journalArticle

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