An Algorithm for Piecewise Linear Approximation of Implicitly Defined Two-Dimensional Surfaces

Let H:RN + 2 → RN be a smooth map and let x₀ ∈ H- 1 (0) be a regular point of H. We present an algorithm for obtaining a global piecewise linear approximation to a component M of the surface H- 1 (0). In particular, for a specific type of triangulation T of RN + 2, we construct a component MT ⊂ HT- 1 (0) where HT is the piecewise linear map that interpolates H on the vertices of T. Error bounds on \|H(x)\|_∞ for x ∈ MT are given in terms of the mesh size of T. Newton-type iterative refinement techniques are given for obtaining points on M starting from points on MT. Sample numerical examples with computer graphical output are given.