The Adaptive Finite Element Technique and Its Synthesis with the Approximately Globally Convergent Numerical Method

Abstract

In Chap. 2, we have described our approximately globally convergent numerical method for a CIP for the hyperbolic $c\,(x)\,{u_{tt}}=\triangle u.$ We remind that the notion of the approximate global convergence was introduced in Definition 1.1.2.1. This method addresses the first central question of this book posed in the beginning of the introductory Chap. 1: Given a CIP, how to obtain a good approximation for the exact solution without an advanced knowledge of a small neighborhood of this solution? Theorems 2.8.2 and 2.9.4 guarantee that, within the frameworks of the first and the second approximate mathematical models respectively (Sects. 2.8.4 and 2.9.2), this approximation is obtained indeed for our CIP.