Source-linked AI summary
Construction of arbitrary order finite element degree-of-freedom maps on polygonal and polyhedral cell meshes
Matthew W. Scroggs, Jørgen S. Dokken, Chris N. Richardson, Garth N. Wells
TL;DR
Higher-order finite element spaces need compatible DOF ordering and orientations across shared entities, but mesh reordering is nonlocal or impossible for some quadrilateral and hexahedral meshes. The paper instead composes local permutations and sub-entity transformations, supporting arbitrary elements and mixed polytope meshes. The method is implemented in FEniCSx and preserves input mesh ordering while avoiding parallel communication for constructing entity permutation data.
Problem
Higher-order spaces require neighboring cells to agree on shared DOF layouts and vector orientations, while existing reordering approaches are nonlocal for quadrilaterals and fail for some hexahedral meshes.
Method
The method transforms DOFs on each cell sub-entity using permutations and base transformations, without mesh reordering, multiple reference elements, or runtime basis evaluation.
Results
The method applies to arbitrary-degree Ciarlet finite element spaces on triangles, tetrahedra, quadrilaterals, hexahedra, general polytopes, and mixed meshes.
Takeaways & Limitations
Entity permutation data requires only global mesh-vertex numbering, preserving input ordering and avoiding parallel communication while enabling high-degree elements on hexahedral meshes.
Takeaways & Limitations
Facet integrals and vector-valued basis functions still require care because facet orientations are not guaranteed to agree across cells.
Abstract
from arXiv · showhide
We develop a method for generating degree-of-freedom maps for arbitrary order finite element spaces for any cell shape. The approach is based on the composition of permutations and transformations by cell sub-entity. Current approaches to generating degree-of-freedom maps for arbitrary order problems typically rely on a consistent orientation of cell entities that permits the definition of a common local coordinate system on shared edges and faces. However, while orientation of a mesh is straightforward for simplex cells and is a local operation, it is not a strictly local operation for quadrilateral cells and in the case of hexahedral cells not all meshes are orientable. The permutation and transformation approach is developed for a range of element types, including Lagrange, and divergence- and curl-conforming elements, and for a range of cell shapes. The approach is local and can be applied to cells of any shape, including general polytopes and meshes with mixed cell types. A number of examples are presented and the developed approach has been implemented in an open-source finite element library.
1 INTRODUCTION
Higher-order finite element spaces require neighboring cells to agree on shared DOF ordering and, for vector-valued elements, edge and face orientations. Existing reordering and coordinate-based approaches become nonlocal, limited, or inapplicable for some cell types, motivating local sub-entity transformations.
- Motivation: Higher-order conforming spaces require neighboring cells to agree on shared DOF layouts and vector orientations.The issue is straightforward for low-order vertex or single edge/face DOFs but non-trivial at higher order.
- Motivation: Consistent ordering must handle reversed edges and rotated or reflected faces when cells are permuted.Scalar DOFs require matching order, while vector-valued DOFs additionally require matching directions.
- Current approaches: Traditional node-based meshes make DOF maps easy but usually restrict spaces to low-degree Lagrange elements and require a new mesh for degree changes.
- Current approaches: Reordering is local and always possible for simplex meshes, but quadrilateral reordering is nonlocal and some hexahedral meshes are not orientable.Quadrilateral and hexahedral meshes may require multiple reference elements, global reordering, or stored orientation flags.
- Current approaches: Reordering can also lose necessary normal-orientation information for two-dimensional meshes embedded in three dimensions and related coupled problems.Reflected cells reverse normals, so cell reflection status must sometimes be stored explicitly.
- Proposed approach: The proposed approach transforms DOFs on each cell sub-entity instead of reordering cells, supporting arbitrary elements, mixed polytopes, and vector-oriented cases.It composes one edge or two face transformations per sub-entity, reducing stored transformation matrices.
2 DEGREES-OF-FREEDOM
Finite elements assign local degrees of freedom to reference-cell sub-entities, then map them to global mesh degrees of freedom to obtain the desired continuity. The section introduces point evaluations and integral moments, including scalar, normal, and tangential vector-valued examples.
- A Ciarlet finite element consists of a reference element, a finite-dimensional polynomial space, and a basis of its dual space.
- Each local degree of freedom is associated with a reference-cell sub-entity, and shared sub-entity degrees of freedom receive the same global number on neighbouring cells.This assignment enforces the required continuity of the mapped finite element space.
- 2.1 Point evaluations: Point-evaluation degrees of freedom evaluate a function at fixed reference-element points, as in degree 1 Lagrange elements using triangle vertices.The corresponding basis function is one at its associated vertex and zero at the others.
- 2.2 Point evaluations of face tangents: Vector point evaluations additionally specify a direction; face-tangential evaluations commonly collocate two independent degrees of freedom at each point.Shared faces require compatible directions between adjacent cells.
- 2.3 Integral moments: Integral-moment degrees of freedom integrate scalar or vector fields against weight functions over cells, edges, or faces.Examples include Raviart–Thomas edge moments using facet normals and Nédélec edge moments using tangents.
- 2.3 Integral moments: A degree 2 Nédélec first-kind edge element on a tetrahedron uses twenty degrees of freedom, including two integral moments on each edge and additional face moments.
3 PERMUTING AND TRANSFORMING CELL BASIS FUNCTIONS
The method transforms reference-cell basis functions by independently handling degrees of freedom on mesh sub-entities and composing base transformations determined by entity orientations. It supports scalar, divergence-conforming, and curl-conforming spaces across varied cell types.
- The approach defines a cell-specific matrix that transforms reference basis functions after comparing each cell entity with the reference orientation.The reference orientation and entity numbering may be chosen arbitrarily, subject to a stated convention.
- Because degrees of freedom belong to single mesh entities, transformations can be split into independent base transformations for edges and faces.Edges use transformations for reversal, while faces require transformations representing two independent operations.
- Scalar-valued functions: Scalar spaces commonly use permutation matrices, although higher-degree serendipity face transformations on hexahedra can be linear rather than permutations.
- Divergence-conforming spaces: Divergence-conforming spaces correct disagreeing facet normals by multiplying the associated basis functions on one cell by -1.The method reuses entity-reflection information computed while determining the required base transformations.
- Curl-conforming spaces: Curl-conforming spaces require compatible tangential components; shared three-dimensional faces therefore need linear transformations for rotating and reflecting face-tangent degrees of freedom.Two face-tangent degrees of freedom may be collocated because the tangent space has dimension 2.
- The cell transformation is obtained by composing the required base transformations, and it may be applied during assembly or absorbed into scalar degree-of-freedom numbering when the transformations are permutations.
4 EXAMPLES
The examples apply entity-specific permutations and linear transformations to construct DOF maps across simplex, quadrilateral, hexahedral, tetrahedral, and mixed-cell meshes. They show that orientation corrections preserve shared DOF layouts and continuity, while some vector-valued elements require transformations beyond permutations.
- Examples: The examples cover degree 3 Lagrange triangles, degree 2 Raviart–Thomas quadrilaterals, degree 4 Lagrange hexahedra, and degree 2 Nédélec tetrahedra.The section also extends the approach to a mixed mesh containing different cell types.
- Example 1: Lagrange on a triangle: For degree 3 Lagrange elements on triangles, reversing an edge swaps the corresponding interior edge DOFs, so base transformations are permutations.The three edge reversals exchange DOF pairs 3–4, 5–6, and 7–8.
- Example 2: Raviart–Thomas on a quadrilateral: Degree 2 Raviart–Thomas elements require both swapping edge DOFs and multiplying them by -1 because edge reversal also reverses their directions.Consequently, the cell transformations are linear transformations rather than direct DOF-map permutations.
- Example 3: Lagrange on a hexahedron: Degree 4 Lagrange elements on hexahedra require edge reversals plus face rotations and reflections, represented by composed permutation transformations.The face transformations act on the DOFs associated with the affected face.
- Example 4: Nédélec on a tetrahedron: For degree 2 Nédélec elements on tetrahedra, edge reversals change DOF directions, while face rotations and reflections transform the directional face DOFs.The face DOFs are not necessarily moved by permutation; their directions can still change.
- Application of the transformations: Applying the correct cell-specific transformations aligns shared edge DOFs, whereas omitting them leaves local edge positions inconsistent and makes global basis functions discontinuous.The triangular-mesh example compares the resulting DOF numbering with and without permutations.
5 IMPLEMENTATION
The implementation computes element-specific base transformations and applies their compositions during finite element evaluation. Basix derives transformations for integral-moment DOFs through interpolation, while DOLFINx applies the required operations to cell outputs without explicitly forming full matrices.
- Implementation: FEniCSx implements the DOF transformation method across Basix, FFCx, and DOLFINx.Basix stores element transformation data, FFCx evaluates forms on a common reference cell, and DOLFINx applies cell-specific transformations to kernel output.
- Implementation: The implementation composes base transformations directly on reference-basis evaluations instead of explicitly constructing each cell matrix M.Because each base transformation affects only a small subset of basis functions, updates can be restricted to those functions.
- Computing base transformations: For integral-moment DOFs, base transformations are computed from interpolation matrices into the spaces used by the moment functionals.The required interpolation coefficients are generated when an element is initialized and reused for transformation construction.
- Computing base transformations: For a serendipity face moment, a face coordinate transformation such as T: (t,s) ↦ (s, 1 − t) rotates the underlying dPc basis and determines transformed functionals.The transformed basis is interpolated into the face space, allowing the corresponding moment functionals and basis functions to be related.
- Computing base transformations: The same construction extends to face reflections and integral moments on other mesh sub-entities.These transformations are obtained by applying the analogous interpolation-based procedure.
6 CONCLUSIONS
The paper presents an arbitrary-degree, element-specific transformation method that works across cell types without reordering the mesh. Its entity data uses only global vertex numbering, preserving input ordering and avoiding parallel communication.
- Scope: The method constructs arbitrary-degree Ciarlet finite element spaces for triangles, tetrahedra, quadrilaterals, hexahedra, general polygons, polyhedra, and mixed meshes.It supports a range of DOF types and therefore a range of finite element types.
- Transformation types: Some elements use DOF permutations, while others require linear transformations applied to the local element tensor for each cell.The distinction depends on whether orientation effects can be represented by permutations alone.
- Practical implications: The required entity permutation data needs only global mesh vertex numbering, preserving input ordering and requiring no parallel communication.The authors identify this property as particularly useful for high-performance computing with high-degree hexahedral elements.