Overview

In this chapter, we consider mesh construction from the perspective of a finite element style computation. The objective is to introduce some mesh generation methods or mesh modification methods resulting in a mesh that conforms to some pre-specified requirements in terms of sizes (isotropic problem) or in terms of sizes and directional properties (anisotropic problem). The basic principle for adapted mesh construction is to collect information about the sizes, the directions and the related sizes using an adequate data structure, a control space (or a background mesh), and then to use this information to construct a mesh conforming (as far as possible) to these specifications.

As pointed out in Chapter 20 for finite element convergence and accuracy, theoretical error estimates involve the parameter h, the size of the mesh elements. Thus, at least for isotropic situations, the necessity of adapting the hs in the mesh is rather natural ^[1]. On the other hand, anisotropic cases are not so trivial. Actually, numerical experiments (mainly in two dimensions) resulting in nice solutions could lead us to think that h-adaptation is well suited to the problem, and yet theoretical proofs are not fully available at this time (except for simple problems).

Hence, we discuss a mesh adaptation method related to the element size (direction), in other words an h-method. This type of method is, as will be seen, the basic ingredient that makes it possible to compute the solution by means of adaptive solution methods. In brief,...