[Paper Review] A systematic construction of finite element commuting exact sequences

Bernardo Cockburn, Guosheng Fu|arXiv (Cornell University)|Apr 30, 2016

Advanced Numerical Methods in Computational Mathematics13 references3 citations

TL;DR

This paper presents a systematic method for constructing finite element commuting exact sequences on general polyhedral elements in one, two, and three dimensions, ensuring stability in mixed finite element methods. By leveraging polynomial shape functions and recursive dimension-by-dimension construction, it rediscoveries known sequences for standard elements and uncovers new families for polygons, prisms, and pyramids, achieving significant dimension reduction in some cases—particularly for pyramids compared to prior rational-function-based approaches.

ABSTRACT

We present a systematic construction of finite element exact sequences with a commuting diagram for the de Rham complex in one-, two- and three-space dimensions. We apply the construction in two-space dimensions to rediscover two families of exact sequences for triangles and three for squares, and to uncover one new family of exact sequence for squares and two new families of exact sequences for general polygonal elements. We apply the construction in three-space dimensions to rediscover two families of exact sequences for tetrahedra, three for cubes, and one for prisms; and to uncover four new families of exact sequences for pyramids, three for prisms, and one for cubes.

Motivation & Objective

• To develop a general, systematic framework for constructing commuting exact sequences of finite element spaces on arbitrary polyhedral elements in 1D, 2D, and 3D.
• To unify and extend existing constructions for standard elements such as tetrahedra, cubes, prisms, and pyramids by deriving them from a single coherent method.
• To identify and construct new families of exact sequences for general polygonal and polyhedral elements, including previously unknown families for squares, prisms, and pyramids.
• To achieve significant dimension reduction in high-order sequences, particularly for pyramidal elements, compared to existing rational-function-based constructions.
• To provide a practical, constructive alternative to Virtual Element Methods and Finite Element Systems by avoiding solutions to PDEs and enabling explicit, computable basis functions.

Proposed method

• Constructing exact sequences recursively by starting from 1D, extending to 2D polygonal elements, and then to 3D polyhedral elements, ensuring continuity and commutativity across the de Rham complex.
• Using polynomial shape functions in tensor-product and hierarchical constructions to define finite element spaces on reference elements, with careful control of degrees of freedom and polynomial degrees.
• Employing dimension counting and kernel analysis (e.g., showing that ker(∇×) ⊂ ∇H) to verify exactness of the sequences, particularly through decomposition of polynomial spaces in multiple variables.
• Applying the method to standard reference elements (tetrahedra, cubes, prisms, pyramids) to recover known sequences and to general elements to discover new ones.
• Utilizing tensor-product and mixed polynomial spaces (e.g., P_k|k, P_k|k+1, RT_k) to build compatible spaces across different differential forms in the de Rham complex.
• Verifying the commuting diagram property by proving that the interpolation operators commute with gradient, curl, and divergence operators via explicit polynomial analysis and coefficient matching.

Experimental results

Research questions

• RQ1 Can a unified, systematic construction method be developed for finite element commuting exact sequences across arbitrary polyhedral elements in multiple dimensions?
• RQ2 Which known families of finite element sequences for standard elements (e.g., tetrahedra, cubes, prisms) can be recovered using this systematic approach?
• RQ3 What new families of exact sequences exist for general polygonal and polyhedral elements that were previously undiscovered?
• RQ4 How does the dimension of the resulting finite element spaces compare to existing constructions, especially for challenging elements like pyramids?
• RQ5 Can the method produce minimal compatible finite element systems (mcFES) in a practical and constructive way, avoiding reliance on implicit PDE-based basis functions?

Key findings

• The method successfully recovers two families of exact sequences for triangles and three for squares in 2D, and uncovers one new family for squares and two for general polygonal elements.
• In 3D, the construction rediscoveries two families for tetrahedra, three for cubes, and one for prisms, and uncovers four new families for pyramids, three for prisms, and one for cubes.
• For pyramidal elements, the new construction achieves a significant dimension reduction compared to the rational-function-based sequence in [22,23], resulting in a more efficient and practical finite element space.
• The exactness of the sequences is rigorously proven by showing that the kernel of the curl operator on the vector-valued space is exactly the image of the gradient operator on the scalar space, using polynomial decomposition and coefficient analysis.
• The dimension formulas for the spaces in the sequences are explicitly derived and verified, with precise counts such as dim H₃ = 3(k+1)(k+2)(k+3)/2 for the 2D sequence S_{6,k}^{2d}.
• The construction provides a practical, explicit method for building minimal compatible finite element systems (mcFES) without requiring solution of PDEs or complex harmonic form subsystems, offering a constructive alternative to existing FES and VEM frameworks.