Geometric decompositions of the simplicial lattice and smooth finite elements in arbitrary dimension

TL;DR

This paper introduces a geometric decomposition method for simplicial lattices that simplifies the construction and implementation of $C^m$-conforming finite elements in arbitrary dimensions.

Contribution

It presents a new geometric approach to decompose the simplicial lattice, enhancing the construction and implementation of smooth finite elements.

Findings

Simplifies the construction of finite elements using geometric decomposition.

Provides an easy implementation method based on graph distance.

Extends the approach to arbitrary dimensions.

Abstract

Recently $C^m$-conforming finite elements on simplexes in arbitrary dimension are constructed by Hu, Lin and Wu. The key in the construction is a non-overlapping decomposition of the simplicial lattice in which each component will be used to determine the normal derivatives at each lower dimensional sub-simplex. A geometric approach is proposed in this paper and a geometric decomposition of the finite element spaces is given. Our geometric decomposition using the graph distance not only simplifies the construction but also provides an easy way of implementation.