Orientation embedded high order shape functions for the exact sequence elements of all shapes

In the context of finite elements, construction of higher order shape functions for elements forming the exact sequence has been a long standing activity in both engineering and numerical analysis communities. A comprehensive review of the subject can be found for example in Demkowicz (2006), Demkowicz et al. (2007) and references therein. This document presents a self-contained systematic theory for the construction of a particular set of hierarchical, orientation embedded, H 1 , H(curl), H(div), and L 2 conforming shape functions for elements of " all shapes " , forming the 1D, 2D, and 3D commuting exact sequences discussed within. By elements of " all shapes " , we specifically mean the segment (unit interval) in 1D, the quadrilateral and triangle in 2D, and the hexahedron, tetrahedron, prism (wedge) and pyramid in 3D. Triangle Segment Figure 1.1: Elements of " all shapes ". There are many ways to construct sets of shape functions satisfying the aforementioned properties. However, we believe that in this work we have constructed a set which strikes an uncommon balance between simplicity and applicability. For all elements, and each associated energy space, we rely upon a simple methodology and a very small collection of ancillary functions to generate all of our shape functions. Furthermore, we have supplemented this text with a package written in Fortran 90 defining each function presented in this work. 1 For these reasons, when reproducing our work in their own software, the readers should find the burden of implementation minimal. We hope that our exposition will be clear and useful, particularly to those less familiar with the subject. For those at the forefront of shape function construction, we hope that our work will be intriguing if only for the elegance of our construction. Particularly, we evidence §9 on pyramid shape functions. The higher order discrete commuting exact sequence for this element appeared only recently in the work of Nigam and Phillips (2012). Our construction for the pyramid presents shape functions spanning each of their discrete energy spaces while maintaining compatibility with the other 3D elements. We also remark that, for any given mesh, our shape functions are fully compatible across adjacent interelement boundaries due to considering so-called orientation embeddings. Hence, no alterations of the shape functions are necessary at the finite element assembly procedure. Moreover, these orientation embeddings are handled almost effortlessly by simply permuting the entries of a few relevant functions. …