DefElement Builds a Verified Encyclopedia of Finite Element Definitions

A team of computational mathematicians has unveiled a systematic effort to bring order to one of numerical analysis’s most sprawling corners: the definitions of finite elements. The project, called DefElement, is an online encyclopedia hosted at defelement.org that catalogues the definitions of a large number of finite elements in a single, standardised format. Created and maintained by the authors of a new open-access paper published in the Springer journal Computational Science and Engineering on 14 April 2026, the encyclopedia aims to solve a problem that has quietly plagued the finite element community for decades: the same element may be described differently in different papers, implemented differently in different software libraries, and it is genuinely hard to tell whether two implementations are really the same mathematical object.

Finite elements are the building blocks of the finite element method, a technique with origins in the 1940s when engineers began solving continuum problems by meshing a domain into small pieces. The method got its name in a 1960 paper by Ray Clough, and its modern formal foundation arrived in 1975, when Philippe Ciarlet introduced the idea of representing a finite element as a triple in lecture notes, later popularised in his 1978 book. That triple consists of a reference cell, a finite-dimensional function space on it, and a set of linear functionals forming a basis of the dual space. The triple was quickly extended by Raviart and Thomas and by Nédélec to vector-valued elements conforming in the H(div) and H(curl) function spaces, and it remains the widely accepted definition of a finite element today.

The practical difficulty DefElement addresses is fragmentation. Often only the lowest-order version of an element on a single cell type is introduced in the original paper, with higher-order versions and versions on other cell types appearing in later publications. Tracking down the right element for a particular problem can therefore be a challenge. Each DefElement page, illustrated in the paper with the Raviart–Thomas element, lists alternative names, the full definition, the polynomial and Lagrange subdegree and superdegree, the number of degrees of freedom as a function of the degree k, details of any variants, the libraries in which the element is implemented, example low-degree elements with plots generated using Symfem, and references to the literature where the full definition can be found. All diagrams are licensed under a Creative Commons license so they can be reused with attribution.

A distinctive feature of the encyclopedia is that it does not merely describe elements; it checks them. Because many open-source finite element libraries exist, including Basix, Bempp-cl, FIAT, ndelement and Symfem, it can be difficult to verify that an implementation matches the element’s definition in the literature or in another library, especially when libraries include variants whose basis functions do not match exactly. The new paper carefully derives conditions under which two elements can be considered equivalent and describes an algorithm that uses these conditions to verify that two implementations are indeed variants of the same element. The results of scheduled runs of this verification algorithm are published on the DefElement website, shown next to each library’s name as a green tick or an orange or red cross.

The mathematical core of the verification is the notion of a variant. Two reference-mapped finite elements are variants of each other if they share the same reference cell and the same polynomial space, and if, for every sub-entity of the cell, the sets of functionals associated with that sub-entity are equivalent. Equivalence here means the two sets have the same size and there exists a push-forward map under which each functional reproduces the Kronecker delta pattern on the corresponding basis functions. Crucially, variants can have different basis functions while sharing the properties that determine the accuracy of a computed solution, so verification does not demand identical basis functions across libraries.

To make this checkable from basis functions alone, the authors introduce the uncontrolled and controlled trace spaces of an element’s polynomial space on a sub-entity. The uncontrolled trace space contains the restrictions to the sub-entity of functions that vanish under every functional associated with the closure of that sub-entity; the controlled trace space is its orthogonal complement within the restricted polynomial space. For a Lagrange element, the uncontrolled trace space is trivial, since point evaluations pin down everything on the sub-entity. For a degree 1 Raviart–Thomas element on a triangle, by contrast, the controlled trace space on an edge contains exactly the normal components that are continuous between cells, while the uncontrolled space holds the tangential parts that may jump. The paper’s central theorem proves that if two elements with the same reference cell and polynomial space have equal uncontrolled trace spaces on every sub-entity, then they are variants of each other.

The verification algorithm builds directly on this theorem. It first checks that the two implementations use the same reference cell, applying a push-forward map if not, then checks that the polynomial spaces coincide, that the same number of degrees of freedom are associated with each sub-entity, and finally that the uncontrolled trace spaces match for every sub-entity. Because spaces cannot be compared symbolically in software, the implementation evaluates candidate basis functions on a regular lattice of points dense enough to capture the highest-degree polynomial being verified, forms matrices from the evaluated values, and confirms by rank checks that the row spans of the two matrices are identical. Verification is performed monthly, currently for elements implemented in Basix, FIAT and ndelement, with Symfem serving as the baseline. According to the authors, this process has already led them to find and correct a number of bugs in FIAT and Symfem.

The encyclopedia also documents the theoretical landscape in which these elements live. Its reference material covers scalar-valued elements for H¹ and L² spaces, including Lagrange, serendipity, Argyris, Hermite and Crouzeix–Raviart elements; vector-valued elements for H(div) and H(curl), such as Raviart–Thomas, Brezzi–Douglas–Marini and both Nédélec families; matrix-valued elements for the H(div div), H(curl curl) and H(curl div) spaces, including Regge and Hellan–Herrmann–Johnson elements; macroelements built on subdivided cells, such as P1-iso-P2, Hsieh–Clough–Tocher and Guzmán–Neilan elements; and dual elements on the barycentric dual grid used in boundary element preconditioning. It explains the push-forward maps, including the contravariant and covariant Piola maps, that preserve the continuity properties each element type requires, and it situates families of elements within discrete de Rham complexes, whose structure is often key to stable and accurate discretisations.

One subtle contribution is a careful treatment of element degree. On simplex cells, the polynomial degree and the degree of the Lagrange element spanning the natural polynomial space coincide, but on other cell types, such as quadrilaterals, hexahedra and pyramids, they can disagree. The paper defines polynomial and Lagrange subdegrees and superdegrees, notes that some measures behave poorly, for example serendipity elements always have Lagrange subdegree 1 on a quadrilateral, and explains that DefElement generally indexes elements by polynomial subdegree. The authors point out that this indexing differs from that used by some libraries, including FEniCS, for elements such as Nédélec first kind and Raviart–Thomas, a discrepancy the encyclopedia makes explicit rather than hiding.

The entire project is open. Each element on DefElement is defined by a YAML-format .def file, the full site is generated by Python scripts, and the code is available under an MIT license on GitHub, with three-monthly snapshots archived on Zenodo. The authors act as editors, checking contributions through pull request reviews, and they encourage readers to add new elements, report errors, or register their own finite element libraries; libraries with a Python interface can implement a verification function so that their elements are automatically checked each month. As the number of finite elements in the literature continues to grow, the team argues, concise and consistent information about them, together with routine verification that implementations match their definitions, is increasingly important, and DefElement is designed to provide exactly that.

Subject of Research: An online encyclopedia of finite element definitions with an automated algorithm verifying that implementations in open-source libraries are variants of the same element

Article Title: DefElement: an encyclopedia of finite element definitions

Article References: Scroggs, M. W., Brubeck, P. D., Dean, J. P., Dokken, J. S., & Marsden, I. (2026). DefElement: an encyclopedia of finite element definitions. Computational Science and Engineering, 3(1), Article 2. https://doi.org/10.1007/s44207-026-00011-0

Image Credits: AI Generated

DOI: 10.1007/s44207-026-00011-0

Keywords: finite element method, DefElement, numerical analysis, open-source software, element verification, Ciarlet triple, H(div) conformity, H(curl) conformity, polynomial bases, Symfem, Basix, computational mathematics