The approximation of generalized turning points by projection methods with superconvergence to the critical parameter

Abstract

Summary.A procedure is given that generates characterizations of singular manifolds for mildly nonlinear mappings between Banach spaces. This characterization is used to develop a method for determining generalized turning points by using projection methods as a discretization. Applications are given to parameter dependent two-point boundary value problems. In particular, collocation at Gauss points is shown to achieve superconvergence in approximating the parameter at simple turning points.

The approximation of generalized turning points\newline by Krylov subspace methods

“…The knowledge of these eigenvectors is important for moving from the singular point along solution paths of (1.2) (see e. g. [7]). Moreover, as pointed out in [6], the accuracy of the approximations of x * , as solution of (1.4)-(1.5), is in direct dependence on the accuracy of the approximations of the y * i 's.…”

“…The type of singular solutions of (1.2) we take here into consideration are the so-called generalized turning points, which Griewank and Reddien characterized in [5,6] as regular solutions of suitable larger systems. Such points comprehend various singularities like turning points and, after unfolding, bifurcation points and isolas.…”

“…Although here we will refer to the general characterization of Griewank and Reddien, it is not difficult to realize how our approximation method can be extended also to other types of augmented systems related to critical points. We refer to [6] for a discussion on various characterizing systems given in the literature.…”

Numerical computation of symmetry-breaking bifurcation points

“…are normalisations for v and u respectively, which forces f x to be singular, see Attili [1,2] and Griewank and Reddien [4,5]. Also v and R are chosen to be in R n a .…”

“…To compute such points, we use the extended system proposed by Griewank and Reddien [4,5]; that is, where g(x, k) = u J f x (x, k) • v and u and v are the left and right null vectors of /JJ 0 respectively. More on this system will be given in Section 3 (see also Attili [1,2]).…”