PL Continuation Methods

Abstract

In previous chapters we assumed that the map H : R^N+1 → R^N was smooth, that zero was a regular value, and that H⁻¹ (0) 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 H _T which affinely interpolates H at the nodes of a triangulation T of R^N+1. The PL methods trace the piecewise linear 1-manifold H _T ⁻¹ (0). 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.