Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes

Andrew Gillette, Alexander Rand, Chandrajit Bajaj

Research output: Contribution to journalArticle

15 Citations (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)667-683
Number of pages17
JournalComputational Methods in Applied Mathematics
Volume16
Issue number4
DOIs
StatePublished - Oct 1 2016

Fingerprint

Basis Functions
Scalar
Mesh
Finite Element
Calculus
Barycentric Coordinates
Convex Polytope
Convex polyhedron
Convex polygon
Vector-valued Functions
Meshing
Differential Forms
Polytope
Function Space
Lowest
Count
Finite Element Method
Generalise
Polynomial
Polynomials

Keywords

  • Finite Element Exterior Calculus
  • Generalized Barycentric Coordinates
  • Polygonal Finite Element Methods

ASJC Scopus subject areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

Cite this

Construction of Scalar and Vector Finite Element Families on Polygonal and Polyhedral Meshes. / Gillette, Andrew; Rand, Alexander; Bajaj, Chandrajit.

In: Computational Methods in Applied Mathematics, Vol. 16, No. 4, 01.10.2016, p. 667-683.

Research output: Contribution to journalArticle

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