12. PL Continuation Methods
Society for Industrial and Applied Mathematics eBooks · 2003
12.1 Introduction In previous chapters we assumed that the map was smooth, that zero was a regular value, and that was a collection of disjoint smooth curves which could be numerically traced using PC-methods. Now we will discuss piecewise linear (PL) methods which can again be viewed as curve tracing methods, but the map H can now be arbitrary. The map H is approximated by a piecewise linear map which affinely interpolates H at the nodes of a triangulation of . The PL methods trace the piecewise linear 1-manifold . A connected component of the piecewise linear 1-manifold consists of a polygonal path which is obtained by successively stepping through certain “transverse” (N + 1)-dimensional simplices of the triangulation. Although the PL method works for arbitrary maps, only under some smoothness assumptions on H can one obtain truncation error estimates in terms of the meshsize of the underlying triangulation. In order to be able to discuss these methods it is necessary to introduce a few combinatorial ideas. The first notions we need are those of a simplex and a triangulation. (12.1.1) Definition. A set of points in is said to be affinely independent (also called in general position) if the following matrix has full rank i.e. if its columns are linearly independent: 11…1 v1 v2 … vk+1 . 12.1.2 Equivalently, are affinely independent if the differences are linearly independent vectors in . Note that one point is always affinely independent. The notion of a simplex is basic for the description of PL methods.