Une méthode numérique pour le calcul des points de retournement. Application à un problème aux limites non-linéaire

SummaryAn iterative method is presented for determining a turning point in a branch of solutions of an algebraic system of equations depending on a parameter. One shows that the convergence is superlinear and that this method may constitute a complement for the Newton method. With a finite element method, numerical results are given testing the process on the discrete version of a mildly nonlinear boundary value problem to determine the turning point in the positive solution branch.

[1]  H. Amann.  Fixed Point Equations and Nonlinear Eigenvalue Problems in Ordered Banach Spaces , 1976 .

[2]  Milan Kubicek,et al.  Algorithm 502: Dependence of Solution of Nonlinear Systems on a Parameter [C5] , 1976, TOMS.

[3]  An Iterative Finite Element Scheme for Bifurcation Analysis of Semi-Linear Elliptic Equations , 1976 .

[4]  James M. Ortega,et al.  Iterative solution of nonlinear equations in several variables , 2014, Computer science and applied mathematics.

[5]  M. Yamaguti,et al.  On numerical deformation of singularities in nonlinear elasticity , 1979 .

[6]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[7]  A numerical method for computing turning points: II. Analyse numrique d'un problme aux limites non-linaire , 1981 .

[8]  R. B. Simpson.  A Method for the Numerical Determination of Bifurcation States of Nonlinear Systems of Equations , 1975 .

[9]  Fulbert Mignot,et al.  Sur une classe de problemes non lineaires avec non linearite positive, croissante, convexe. , 1980 .

[10]  Michael G. Crandall,et al.  Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems , 1975 .

[11]  J. Sprekels.  Exact bounds for the solution branches of nonlinear eigenvalue problems , 1980 .

[12]  S. S. Antman,et al.  Bifurcation theory and nonlinear eigenvalue problems, 1967 , 1968 .

[13]  G. Reddien.  On Newton’s Method for Singular Problems , 1978 .

[14]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .

[15]  Jacques Rappaz,et al.  Finite Dimensional Approximation of Non-Linear Problems .1. Branches of Nonsingular Solutions , 1980 .

[16]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[17]  F. Kikuchi.  Finite element approximations to bifurcation problems of turning point type , 1979 .

[18]  Prolongement analytique d'une branche de solutions d'une équation aux dérivées partielles non 1inéaire paramétrée. Applications , 1979 .

[19]  D. Sattinger.  Topics in stability and bifurcation theory , 1973 .

[20]  R. B. Simpson.  Existence and error estimates for solutions of a discrete analog of nonlinear eigenvalue problems , 1972 .

[21]  F. Brezzi,et al.  Finite dimensional approximation of nonlinear problems , 1981 .