A Pseudo-Arclength Continuation Method for Nonlinear Eigenvalue Problems.

1985 ◽  
Author(s):  
Hans D. Mittelmann
Keyword(s):  
Eigenvalue Problems ◽  
Continuation Method ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Eigenvalue

SYMMETRY REDUCTIONS AND A POSTERIORI FINITE ELEMENT ERROR ESTIMATORS FOR BIFURCATION PROBLEMS

2005 ◽  
Vol 15 (07) ◽  
pp. 2091-2107 ◽  
Author(s):  
C.-S. CHIEN ◽  
B.-W. JENG
Keyword(s):  
Finite Element ◽  
Eigenvalue Problems ◽  
Continuation Method ◽  
Discretization Scheme ◽  
Element Discretization ◽  
Nonlinear Eigenvalue Problems ◽  
Error Estimators ◽  
Symmetry Reductions ◽  
Relaxation Scheme ◽  
Nonlinear Eigenvalue

We discuss efficient continuation algorithms for solving nonlinear eigenvalue problems. First, we exploit the idea of symmetry reductions and discretize the problem on a symmetry cell by the finite element method. Then we incorporate the multigrid V-cycle scheme in the context of continuation method to trace solution branches of the discrete problems, where the preconditioned Lanczos method is used as the relaxation scheme. Next, we apply the symmetry reduction technique to the two-grid finite element discretization scheme [Chien & Jeng, 2005] to solve some nonlinear eigenvalue problems in physical science. The two-grid centered difference discretization scheme described therein was also implemented for comparison. Sample numerical results are reported.

MULTIPLE POSITIVE SOLUTIONS OF NONLINEAR EIGENVALUE PROBLEMS ASSOCIATED TO A CLASS OF p-LAPLACIAN LIKE OPERATORS

2003 ◽  
Vol 05 (05) ◽  
pp. 737-759 ◽  
Author(s):  
NOBUYOSHI FUKAGAI ◽  
KIMIAKI NARUKAWA
Keyword(s):  
Positive Solutions ◽  
Elliptic Problem ◽  
Eigenvalue Problems ◽  
Multiple Positive Solutions ◽  
Combined Effects ◽  
Nonlinear Eigenvalue Problems ◽  
Quasilinear Elliptic Problem ◽  
Quasilinear Elliptic ◽  
Nonlinear Eigenvalue

This paper deals with positive solutions of a class of nonlinear eigenvalue problems. For a quasilinear elliptic problem (#) - div (ϕ(|∇u|)∇u) = λf(x,u) in Ω, u = 0 on ∂Ω, we assume asymptotic conditions on ϕ and f such as ϕ(t) ~ tp0-2, f(x,t) ~ tq0-1as t → +0 and ϕ(t) ~ tp1-2, f(x,t) ~ tq1-1as t → ∞. The combined effects of sub-nonlinearity (p0> q0) and super-nonlinearity (p1< q1) with the subcritical term f(x,u) imply the existence of at least two positive solutions of (#) for 0 < λ < Λ.

Sign in / Sign up

Export Citation Format

Share Document