title: CONSTRUCTION OF SCALAR AND VECTOR FINITE ELEMENT FAMILIES ON POLYGONAL AND POLYHEDRAL MESHES.
abstract: We combine theoretical results from polytope domain meshing, generalized barycentric coordinates, and finite element exterior calculus to construct scalar- and vector-valued basis functions for conforming finite element methods on generic convex polytope meshes in dimensions 2 and 3. Our construction recovers well-known bases for the lowest order Nédélec, Raviart-Thomas, and Brezzi-Douglas-Marini elements on simplicial meshes and generalizes the notion of Whitney forms to non-simplicial convex polygons and polyhedra. We show that our basis functions lie in the correct function space with regards to global continuity and that they reproduce the requisite polynomial differential forms described by finite element exterior calculus. We present a method to count the number of basis functions required to ensure these two key properties.
authors.affiliation: Department of Mathematics, University of Arizona, Tucson, AZ, USA, agillette@math.arizona.edu.
authors.forename: Andrew
authors.initials: A
authors.lastname: Gillette
authors.affiliation: CD-adapco, Austin, TX, USA, alexander.rand@cd-adapco.com.
authors.forename: Alexander
authors.initials: A
authors.lastname: Rand
authors.affiliation: Department of Computer Science, Institute for Computational Engineering and Sciences, University of Texas at Austin, Austin, TX, USA, bajaj@cs.utexas.edu.
authors.forename: Chandrajit
authors.initials: C
authors.lastname: Bajaj