Orientation Embedded High Order Shape Functions for the Exact Sequence Elements of All Shapes

TL;DR

This paper introduces a unified approach to constructing high order shape functions for all classical energy spaces and element shapes, ensuring conformity, hierarchy, and compatibility for hybrid meshes with local adaptivity.

Contribution

It provides a novel, orientation-embedded, coordinate-free formulation of shape functions that satisfy the exact sequence property for all element shapes and energy spaces.

Findings

Shape functions are conforming, hierarchical, and compatible across shared boundaries.

Expressions are coordinate-free and adaptable to local polynomial orders.

The method supports hybrid meshes with orientation embedding.

Abstract

A unified construction of high order shape functions is given for all four classical energy spaces ($H^1$, $H(\mathrm{curl})$, $H(\mathrm{div})$ and $L^2$) and for elements of "all" shapes (segment, quadrilateral, triangle, hexahedron, tetrahedron, triangular prism and pyramid). The discrete spaces spanned by the shape functions satisfy the commuting exact sequence property for each element. The shape functions are conforming, hierarchical and compatible with other neighboring elements across shared boundaries so they may be used in hybrid meshes. Expressions for the shape functions are given in coordinate free format in terms of the relevant affine coordinates of each element shape. The polynomial order is allowed to differ for each separate topological entity (vertex, edge, face or interior) in the mesh, so the shape functions can be used to implement local $p$ adaptive finite element methods. Each topological entity may have its own orientation, and the shape functions can have that orientation embedded by a simple permutation of arguments.