Raviart–Thomas Finite Elements

• Raviart–Thomas spaces are finite element spaces designed for H(div)-conforming vector fields, ensuring exact local mass conservation and stable, optimal-order error estimates.
• They are constructed via a local polynomial-plus-linear formulation with face and cell moment degrees of freedom that enforce inter-element normal continuity.
• The use of the Piola transformation and commuting divergence properties makes these spaces robust for mixed formulations on complex and polytopal meshes.

The Raviart–Thomas (RT) spaces are a cornerstone in the finite element discretization of $H(\mathrm{div})$-conforming vector fields, widely used in mixed formulations of second-order elliptic partial differential equations. These spaces are defined by a local polynomial-plus-linear structure, admit a canonical Piola transformation to physical domains, enforce inter-element normal-flux continuity via carefully chosen degrees of freedom, and are compatible with the structure of the de Rham complex in finite element exterior calculus. Their construction ensures exact local mass conservation, surjective divergence mapping onto polynomials, and stable optimal-order approximation, even on general polytopal and locally refined meshes.

1. Construction and Algebraic Structure

Let $K\subset\mathbb{R}^d$ be a cell (simplex, parallelepiped, or more generally a polygon/polyhedron). For any degree $k \geq 0$, the local Raviart–Thomas space is defined as

$$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$

Here, $P_k(K)$ denotes polynomials of degree at most $k$ on $K$; $[P_k(K)]^d$ are vector-valued polynomials.

The key properties are:

• For all $v \in \mathrm{RT}_k(K)$, $\mathrm{div}\,v \in P_k(K)$, and $K\subset\mathbb{R}^d$0 (Christiansen et al., 2013, Cruz, 2022).
• On simplicial meshes, $K\subset\mathbb{R}^d$1 can be identified as the $K\subset\mathbb{R}^d$2-form spaces in the trimmed (minus) de Rham sequence within the Arnold-Falk-Winther FEEC framework (Christiansen et al., 2013).

The global space is assembled by

$K\subset\mathbb{R}^d$3

2. Degrees of Freedom and Basis Construction

Degrees of freedom (DOFs) are defined as:

• Face (boundary) moments: For every face $K\subset\mathbb{R}^d$4 and every $K\subset\mathbb{R}^d$5,

$K\subset\mathbb{R}^d$6

• Cell (interior) moments: For every $K\subset\mathbb{R}^d$7,

$K\subset\mathbb{R}^d$8

These DOFs ensure unisolvence—DOFs uniquely determine a function in $K\subset\mathbb{R}^d$9 (Christiansen et al., 2013, Abgrall et al., 2019).

A canonical basis is built so that each function either has a single nonzero face moment (face basis) or vanishes on the boundary (bubble/interior basis). For $k \geq 0$0, there is one DOF per face and the basis functions are affine maps tied to the normal of each face (Ji, 2021).

The RT interpolator $k \geq 0$1 matches these DOFs elementwise and commutes with divergence: $k \geq 0$2 where $k \geq 0$3 is the $k \geq 0$4-projection onto $k \geq 0$5 polynomials (Bartels et al., 2020).

3. Mapping to Physical Elements: Piola Transformation

The contravariant (Piola) transform ensures preservation of divergence and normal traces under affine maps. If $k \geq 0$6 is affine with Jacobian $k \geq 0$7,

$k \geq 0$8

for $k \geq 0$9 and $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$0. This transformation preserves $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$1-conformity and the action of the divergence operator up to the appropriate Jacobian factors (Cruz, 2022, Hu et al., 2014).

4. Conformity, Commuting Diagram, and FEEC Structure

RT spaces are $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$2-conforming: normal components are continuous across faces due to the single-valuedness of the face DOFs. On general polyhedral meshes, polytopal extensions conserve this structure and reduce to classical RT on faces (Abgrall et al., 2019, Abgrall et al., 2019).

Fundamentally, $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$3 fits in the FEEC de Rham sequence as the $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$4-form (trimmed) polynomial space: $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$5 (Christiansen et al., 2013, Berchenko-Kogan, 2021).

The RT interpolator commutes with divergence—i.e., interpolation and divergence commute with $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$6-projection—crucial for preservation of mass conservation and for the inf-sup (Ladyzhenskaya–Babuška–Brezzi) stability (Cruz, 2022, Ji, 2021).

5. Implementation and Variants

Efficient implementations, e.g., in MFEM, exploit symbolic or orthonormal polynomial bases and automate Piola mapping (Cruz, 2022). On general meshes, the basis is constructed by orthonormalizing vector polynomials using, e.g., Dubiner basis and Gram–Schmidt process (Anantharamu et al., 2023). Algorithmic advancements such as the elimination of internal flux subspaces via stabilization (subspace-to-stabilization) reduce computational cost by 10–20% over a large $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$7 range (Anantharamu et al., 2023).

Variants include:

• Immersed RT spaces for unfitted interface problems, modifying basis and DOFs to respect interface discontinuities while keeping optimal approximation and mass conservation (Ji, 2021).
• Hybridized and HDG-like methods via RT subspace decomposition and stabilization (Anantharamu et al., 2023).
• Enriched Scott–Vogelius pairs by RT bubble enrichment that yield pressure-robust, inf-sup stable $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$8 discretizations (John et al., 2022).
• Petrov–Galerkin and dual RT bases for cell-centered finite volume schemes via biorthogonalization (Dubois, 2010).

6. Approximation Properties and Error Analysis

Standard RT mixed methods yield, under regularity $$\mathrm{RT}_k(K) = [P_k(K)]^d \oplus x\,P_k(K) = \{\,\mathbf{p}(x) + x\,q(x) : \mathbf{p} \in [P_k(K)]^d,\ q \in P_k(K)\,\}.$$9, $P_k(K)$0,

$P_k(K)$1

The order is preserved on straight-edged approximations of curved domains—via suitable Petrov–Galerkin variants—or on arbitrary polygons via VEM-inspired construction, under mesh regularity hypotheses (Cruz, 2022, Abgrall et al., 2019, Bertrand et al., 2023).

Immersed RT methods achieve optimal error bounds even on unfitted interface meshes, as long as the shape-regularity and maximal angle conditions hold (Ji, 2021).

7. Extensions: Polytopal, High Order, and Symmetry

RT extensions to arbitrary polygons and polyhedra are built by defining DOFs through traces and moments, and by building the local shape function space from boundary data and harmonic internal constraints ("VEM–RT" spaces) (Abgrall et al., 2019, Abgrall et al., 2019). Canonical constructions via FEEC yield bases in terms of polynomial multiples of Whitney forms, providing representation-theoretic insight and resolutions compatible with symmetry (vertex-invariance), with explicit criteria for when such symmetric bases exist (Christiansen et al., 2013, Berchenko-Kogan, 2021).

These spaces are instrumental for hybridization, local post-processing, robust $P_k(K)$2-commuting projections, and for constructing pressure-robust velocity fields in incompressible flow simulations (John et al., 2022).

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Table: RT Properties Across Key Dimensions

Property	Simplex/Polytope Spaces	Implementation Feature
Structure	$P_k(K)$3	Orthonormalizing Dubiner/monomial
DOFs	Face $P_k(K)$4, cell $P_k(K)$5	Moment-integrals (face & cell)
Conformity	$P_k(K)$6, normal continuity	Enforced by global assembly
Divergence	$P_k(K)$7 surjective	Commuting RT interpolation
Piola mapping	Contravariant, preserves $P_k(K)$8	Internal in MFEM
Variant examples	Immersed, hybridized, VEM-RT	Petrov–Galerkin, stabilization
Error	$P_k(K)$9 optimal	Confirmed by numerical benchmarks

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• Canonical construction, FEEC: (Christiansen et al., 2013, Berchenko-Kogan, 2021)
• Implementation, standard properties: (Cruz, 2022, Ji, 2021, Anantharamu et al., 2023)
• Polytopal, VEM-type extensions: (Abgrall et al., 2019, Abgrall et al., 2019)
• Hybridized and efficient formulation: (Anantharamu et al., 2023)
• Enrichment and stabilized mixed methods: (John et al., 2022)
• Dual basis and finite volume equivalence: (Dubois, 2010)
• Petrov–Galerkin boundary/curved geometry variant: (Bertrand et al., 2023)
• Immersed RT for interfaces: (Ji, 2021)
• Orthogonality and relations with Crouzeix–Raviart: (Bartels et al., 2020, Hu et al., 2014)