[Paper Review] Construction of scalar and vector finite element families on polygonal and polyhedral meshes

Andrew Gillette, Alexander Rand+1 more|arXiv (Cornell University)|May 27, 2014

Advanced Numerical Methods in Computational Mathematics4 citations

TL;DR

This paper constructs conforming scalar and vector finite element bases on general convex polygonal and polyhedral meshes using generalized barycentric coordinates, extending classical Nédélec, Raviart-Thomas, and Brezzi-Douglas-Marini elements beyond simplicial meshes. The method ensures correct global continuity and polynomial reproduction of differential forms, generalizing Whitney forms to non-simplicial elements with provable convergence properties.

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 N\\'ed\\'elec, 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.

Motivation & Objective

• To develop conforming finite element methods on unstructured polygonal and polyhedral meshes that generalize classical finite elements beyond simplicial elements.
• To extend the theory of finite element exterior calculus to non-simplicial convex polytopes using generalized barycentric coordinates.
• To construct local basis functions that preserve global continuity and reproduce the required polynomial differential forms for $\mathcal{P}_1^{-}\Lambda^k$ and $\mathcal{P}_1\Lambda^k$ spaces.
• To provide a systematic method for counting basis functions needed to satisfy continuity and polynomial reproduction requirements on polytope meshes.
• To enable practical finite element methods on complex meshes by identifying reducible basis components that do not affect inter-element continuity.

Proposed method

• Uses generalized barycentric coordinates—Wachspress, Sibson, harmonic, and mean value—on convex polytopes to define local basis functions for scalar and vector finite elements.
• Constructs basis functions as products of barycentric coordinates and their gradients or curls, ensuring compatibility with differential forms in finite element exterior calculus.
• Applies theoretical results from polytope meshing and generalized barycentric coordinates to ensure that the resulting basis functions lie in the correct function spaces ($H^1$, $H(\text{curl})$, $H(\text{div})$).
• Derives explicit formulas for the number of basis functions required for each differential form degree $k$ and mesh dimension $n=2,3$, distinguishing between total, boundary, and polynomial-reproducing components.
• Introduces a counting procedure based on vertex, edge, and face counts to quantify basis size and identify redundant functions that do not contribute to inter-element continuity.
• Validates the construction by proving that the basis functions reproduce the $\mathcal{P}_1^{-}\Lambda^k$ and $\mathcal{P}_1\Lambda^k$ polynomial differential forms on polytope meshes.

Experimental results

Research questions

• RQ1 Can generalized barycentric coordinates be used to construct conforming finite element bases for scalar and vector-valued problems on arbitrary convex polygonal and polyhedral meshes?
• RQ2 Do the resulting basis functions maintain the correct global continuity properties ($H^1$, $H(\text{curl})$, $H(\text{div})$) on non-simplicial elements?
• RQ3 Can the basis functions reproduce the polynomial differential forms of the $\mathcal{P}_1^{-}\Lambda^k$ and $\mathcal{P}_1\Lambda^k$ spaces on polytope meshes?
• RQ4 What is the minimal number of basis functions required to satisfy continuity and polynomial reproduction, and how can redundancy be identified?
• RQ5 How does the basis size scale with mesh complexity, and in which cases can significant reductions be made without compromising continuity?

Key findings

• The construction generalizes classical lowest-order finite elements (Nédélec, Raviart-Thomas, BDM) to non-simplicial convex polygons and polyhedra using generalized barycentric coordinates.
• Basis functions for $\mathcal{P}_1^{-}\Lambda^k$ and $\mathcal{P}_1\Lambda^k$ spaces are constructed and proven to lie in the correct function spaces with required global continuity.
• For a hexahedral element in 3D, the $\mathcal{P}_1\Lambda^1$ basis contains 56 functions, but only 48 are needed for inter-element continuity, indicating potential for basis reduction.
• In the $k=2$ case on polygons and polyhedra, no inter-element continuity is required, so the full basis size is not needed, and discontinuous Galerkin methods may be more practical.
• The $k=0$ basis cannot be reduced, as every basis function $\lambda_i$ contributes to $H^1$-continuity, but $k=1$ and $k=2$ cases on non-simplicial elements allow for significant basis size reduction.
• The method provides explicit formulas for counting basis functions by vertex, edge, and face contributions, enabling systematic analysis of basis size and redundancy.