Local interpolation on nonaffine meshes

In this chapter, we extend the results of Chapter 11 to nonaffine meshes. For simplicity, the functional transformation is the pullback by the geometric mapping, but this mapping is now nonaffine. The first difficulty consists of comparing Sobolev norms. This is not a trivial task since the chain rule involves higher-order derivatives of the geometric mapping. The second difficulty is to define a notion of shape-regularity for mesh sequences built using nonaffine geometric mappings. We show how to do this using a perturbation theory, and we present various examples.