Local interpolation on affine meshes

Abstract

We analyze the local finite element interpolation error for smooth functions. We restrict the material to affine meshes and to relatively simple functional transformations. We introduce the notion of shape-regular families of affine meshes, we study the transformation of Sobolev norms, and we present important approximation results collectively known as the Bramble–Hilbert lemmas. The main result proved in this chapter is an upper bound on the local interpolation error over each mesh cell for smooth functions.