Assembling Adjacency Relations for the Finite Element Method

This work demonstrates the concept of describing adjacency relations as set-valued mappings, which has already been used in graph algorithm implementations for a long time. The primary goal is to provide a way to express matrix sparsity patterns arising from finite element discretisations on unstructured meshes in a directly implementable manner. For this purpose, a formal definition of these mappings and their transposition and composition operations is presented along with investigations on sparsity. Furthermore, two important variants of sparse matrix patterns required by the finite element and the discontinuous Galerkin methods are derived from the theory of set-valued adjacency relation mappings and expressed as compositions thereof. A generic interface is presented in the second part of this work, along with suggested implementations utilising the C++ standard library components of the required operations which can be easily integrated in existing finite element software packages. Worst-case runtime estimates are presented for these implementations.