Crouzeix–Raviart Finite Element Spaces

• Crouzeix–Raviart spaces are nonconforming finite element spaces that use piecewise polynomial functions with average continuity across element faces.
• They generalize to higher polynomial degrees and dimensions, establishing a direct connection with Raviart–Thomas methods through discrete stress equivalences.
• These spaces are applied to elliptic problems, incompressible flows, and surface PDEs, providing stable discretizations and optimal error convergence.

The Crouzeix–Raviart finite element spaces are nonconforming finite element spaces originally introduced for the discretization of second-order elliptic and incompressible flow problems on simplicial meshes. They play a central role in the finite element method, bridging the gap between conforming Galerkin methods and mixed or discontinuous Galerkin approaches. Crouzeix–Raviart (CR) spaces, their enriched variants (ECR), their generalizations to higher polynomial degree and dimension, and their deep connection with mixed finite element spaces such as Raviart–Thomas, are foundational to both the theory and practice of nonconforming discretization.

1. Canonical Definition and Structure

The standard Crouzeix–Raviart space on a simplicial mesh is defined by imposing piecewise polynomial structure with specific continuity constraints on inter-element faces. On a shape-regular triangulation $\mathcal{T}_h$ of a polytopal domain $\Omega\subset\mathbb{R}^d$, the lowest-order ($k=1$) CR space consists of all functions that are affine on each simplex $K\in\mathcal{T}_h$ and satisfy

$$CR_h = \{ v_h\in L^2(\Omega) : v_h|_K\in P_1(K) \ \forall K,\quad \int_F [v_h]\,ds = 0\ \forall F\in\mathcal{F}_h^i \},$$

where $[v_h]_F$ denotes the jump across face $F$ and $\mathcal{F}_h^i$ is the set of interior faces. The degrees of freedom are given by face averages or, equivalently, by the values at face midpoints (Jin et al., 7 May 2025).

The nonconforming nature of CR is characterized by average continuity on faces, not pointwise continuity at vertices. The broken gradient $\nabla_h$ is taken piecewise, and the global function is discontinuous at vertices and along faces except in the average sense.

2. Generalizations: Higher Order and Multidimensional CR Spaces

Crouzeix–Raviart spaces admit systematic generalization to arbitrary polynomial degree $k$ and spatial dimension $\Omega\subset\mathbb{R}^d$0. On a simplicial mesh $\Omega\subset\mathbb{R}^d$1 in $\Omega\subset\mathbb{R}^d$2, the degree $\Omega\subset\mathbb{R}^d$3 CR space may be described as (Bohne et al., 2024)

$\Omega\subset\mathbb{R}^d$4

i.e., the jump across any interior facet is $\Omega\subset\mathbb{R}^d$5-orthogonal to all polynomials of degree $\Omega\subset\mathbb{R}^d$6 on that facet. For $\Omega\subset\mathbb{R}^d$7 even, the canonical basis includes a conforming subspace and one bubble per simplex; for $\Omega\subset\mathbb{R}^d$8 odd, one bubble per interior facet. Functionals dual to the nonconforming basis are given by simplex and facet moments, with a full edge-only description available only in $\Omega\subset\mathbb{R}^d$9 or for $k=1$0.

Explicit local bases in 3D involve hierarchically-constructed reflection- and symmetry-type polynomials that enforce the correct jump orthogonality and ensure linear independence and direct sum decomposition with the conforming space (Jr. et al., 2017). The construction extends to $k=1$1 settings and variable polynomial order (Bressan et al., 18 Feb 2025).

3. Enriched Crouzeix–Raviart Elements and Raviart–Thomas Equivalence

The enriched Crouzeix–Raviart (ECR) element augments the standard local $k=1$2 space with a cell bubble (e.g., $k=1$3 relative to the centroid $k=1$4), yielding

$k=1$5

Global ECR functions enforce the same average jump constraints as in standard CR. A principal result is that, for piecewise constant right-hand side data, the ECR method yields exactly the same discrete stress field as the lowest-order Raviart–Thomas (RT$k=1$6) mixed method:

$k=1$7

where $k=1$8 is the elementwise $k=1$9-projection onto piecewise constants. This equivalence extends to the Stokes pseudo-stress formulation and Laplace eigenproblems, with energy-norm and eigenvalue errors coinciding up to higher order terms (Hu et al., 2014). The result allows for direct transfer of a posteriori estimators and stability properties between the two families.

4. Inf-Sup Stability and Approximation Properties

Crouzeix–Raviart spaces, including generalizations to arbitrary degree, have been shown to satisfy Ladyzhenskaya–Babuška–Brezzi (LBB) inf-sup stability when paired with appropriate pressures (typically piecewise $K\in\mathcal{T}_h$0, mean-free). For all odd degrees $K\in\mathcal{T}_h$1 in 2D, the pair $K\in\mathcal{T}_h$2 is uniformly inf-sup stable on any shape-regular mesh with at least one interior vertex, closing the longstanding Crouzeix–Falk conjecture (Carstensen et al., 2021). In 3D, inf-sup stability at $K\in\mathcal{T}_h$3 (quadratic) has been proven with explicit local bubble basis, eliminating spurious pressure modes endemic to the conforming $K\in\mathcal{T}_h$4 pairs (Sauter et al., 2022).

Approximation properties are classical: the local nonconforming interpolation operator $K\in\mathcal{T}_h$5 defined by matching edge or facet moments admits the estimate

$K\in\mathcal{T}_h$6

Discrete compactness and a mesh-dependent Poincaré inequality ensure strong $K\in\mathcal{T}_h$7 convergence for bounded sequences.

5. Applications, Implementation, and Algorithmic Aspects

CR spaces are deployed in a variety of applications: elliptic problems (Poisson), incompressible flow (Stokes, Navier–Stokes), eigenvalue computations, phase-field topology optimization, and surface PDEs. They can be implemented efficiently due to their local structure—basis functions are supported on adjacent elements, and the assembly differs from conforming methods only in global coupling across facets. For enriched variants, static condensation allows local bubble elimination, reducing the solve to a standard nonconforming formulation (Hu et al., 2014).

For even-degree generalizations, a mild DG-type stabilization that penalizes the one-dimensional Legendre component of jumps suffices to restore full order convergence in $K\in\mathcal{T}_h$8, $K\in\mathcal{T}_h$9, and $$CR_h = \{ v_h\in L^2(\Omega) : v_h|_K\in P_1(K) \ \forall K,\quad \int_F [v_h]\,ds = 0\ \forall F\in\mathcal{F}_h^i \},$$0 refinements (Bressan et al., 18 Feb 2025).

A posteriori error estimators, including gradient recovery and postprocessed eigenvalue corrections, have been tailored to CR and ECR spaces. Asymptotically exact estimators have been proved and tested, enabling cubic or better convergence for eigenvalue approximations using only a single solve (Hu et al., 2019).

CR elements also have favorable properties in mixed methods, including optimal local best-approximation for divergence and very general well-posedness results, even for very low-regularity solutions (Barrios et al., 2021).

6. Interplay with Raviart–Thomas Spaces and Discrete Orthogonality

CR and Raviart–Thomas (RT) spaces are linked by deep algebraic and analytic identities. At the lowest order, the discrete stress fields of RT$$CR_h = \{ v_h\in L^2(\Omega) : v_h|_K\in P_1(K) \ \forall K,\quad \int_F [v_h]\,ds = 0\ \forall F\in\mathcal{F}_h^i \},$$1 and enriched CR coincide. More generally, the $$CR_h = \{ v_h\in L^2(\Omega) : v_h|_K\in P_1(K) \ \forall K,\quad \int_F [v_h]\,ds = 0\ \forall F\in\mathcal{F}_h^i \},$$2-projection of RT vector fields is orthogonal to the space of discrete gradients (broken gradients) of CR functions, allowing for discrete convex duality, block-diagonal preconditioners, and explicit postprocessing formulae for dual variables. Surjectivity properties of the projection and direct sum decompositions clarify the algebraic structure and guide both analysis and algorithm design (Bartels et al., 2020).

7. Surface, Manifold, and Advanced Applications

Nonconforming CR elements extend naturally to discretizations on surfaces and manifolds. For example, the surface CR space—where mid-edge continuity is enforced on edge-midpoints that lie on the target surface—allows for efficient, local, and robust discretization of tangential vector-valued Laplacians. Optimal $$CR_h = \{ v_h\in L^2(\Omega) : v_h|_K\in P_1(K) \ \forall K,\quad \int_F [v_h]\,ds = 0\ \forall F\in\mathcal{F}_h^i \},$$3 and $$CR_h = \{ v_h\in L^2(\Omega) : v_h|_K\in P_1(K) \ \forall K,\quad \int_F [v_h]\,ds = 0\ \forall F\in\mathcal{F}_h^i \},$$4 error estimates, as well as explicit control of geometric errors, are available in these settings (Mehlmann, 2023).

The broad range and robustness of Crouzeix–Raviart spaces—across degree, dimension, and applications—underscore their enduring relevance in finite element theory and practice. Their modular connection to conforming and mixed spaces, capacity for hierarchical extension and $$CR_h = \{ v_h\in L^2(\Omega) : v_h|_K\in P_1(K) \ \forall K,\quad \int_F [v_h]\,ds = 0\ \forall F\in\mathcal{F}_h^i \},$$5-adaptivity, and compatibility with advanced error estimation and duality techniques position them as a central tool for the modern analysis of PDEs by nonconforming finite element methods.