We develop a method for generating degree-of-freedom maps for arbitrary order Ciarlet-type finite element spaces for any cell shape. The approach is based on the composition of permutations and transformations by cell sub-entity. Current approaches to generating degree-of-freedom maps for arbitrary order problems typically rely on a consistent orientation of cell entities that permits the definition of a common local coordinate system on shared edges and faces. However, while orientation of a mesh is straightforward for simplex cells and is a local operation, it is not a strictly local operation for quadrilateral cells and, in the case of hexahedral cells, not all meshes are orientable. The permutation and transformation approach is developed for a range of element types, including arbitrary degree Lagrange, serendipity, and divergence- and curl-conforming elements, and for a range of cell shapes. The approach is local and can be applied to cells of any shape, including general polytopes and meshes with mixed cell types. A number of examples are presented and the developed approach has been implemented in open-source libraries.

Operator products occur naturally in a range of regularised boundary integral equation formulations. However, while a Galerkin discretisation only depends on the domain space and the test (or dual) space of the operator, products require a notion of the range. In the boundary element software package Bempp, we have implemented a complete operator algebra that depends on knowledge of the domain, range, and test space. The aim was to develop a way of working with Galerkin operators in boundary element software that is as close to working with the strong form on paper as possible, while hiding the complexities of Galerkin discretisations. In this article, we demonstrate the implementation of this operator algebra and show, using various Laplace and Helmholtz example problems, how it significantly simplifies the definition and solution of a wide range of typical boundary integral equation problems.