Finite element generation

Abstract

We show how to generate a finite element in each mesh cell from a reference finite element. To this purpose, we need one new concept in addition to the geometric mapping: a functional transformation that maps functions defined on the current mesh cell to functions defined on the reference cell. Key examples of such transformations are the Piola transformations. These transformations arise naturally in the chain rule when one investigates how the standard differential operators (gradient, curl, divergence) are transformed by the geometric mapping. The construction presented in this chapter provides the cornerstone for the analysis of the finite element interpolation error.