Crouzeix–Raviart Finite Element Spaces

• Crouzeix–Raviart finite element spaces are nonconforming spaces that enforce weak, facet-based continuity, making them effective for discretizing PDEs.
• They are constructed via local polynomial approximations on mesh simplices with average continuity constraints, leading to explicit basis functions and streamlined DOF interpolation.
• Their strong stability properties, including inf-sup conditions and connections to Raviart–Thomas spaces, support efficient mixed formulations and adaptive hp strategies.

The Crouzeix–Raviart finite element spaces are a fundamental family of nonconforming finite element spaces widely used for the discretization of partial differential equations (PDEs), including the Poisson, Stokes, and plate equations, among others. Nonconforming refers to finite element spaces whose functions do not satisfy global continuity constraints (such as full $H^1$-conformity), but instead obey weaker continuity conditions, typically imposed on average over lower-dimensional interfaces (edges or faces). The Crouzeix–Raviart (CR) approach is notable for offering robust approximation properties, inf-sup stability for saddle-point problems such as Stokes flow, and close connections to mixed finite element methods, notably the Raviart–Thomas (RT) spaces. These spaces have seen substantial generalizations in both order and dimension, and have become a backbone for nonconforming discretizations in computational science.

1. Construction and Algebraic Structure

Crouzeix–Raviart spaces are constructed by specifying, on each mesh simplex (triangle, tetrahedron, or higher-dimensional analogue), a local polynomial space and imposing weak continuity constraints across interelement facets. For a mesh $\mathcal{T}$ of $\Omega\subset\mathbb{R}^d$ into simplices, a general CR space of degree $k$ is defined by

$$\text{CR}_k(\mathcal{T}) := \left\{ v \in L^2(\Omega)\;\middle|\; v|_K \in P_k(K)\ \forall K\in\mathcal{T},\ \int_F [v] q = 0\ \forall q\in V_F \right\},$$

where $[v]$ denotes the jump across the facet $F$, and $V_F$ is a subspace of polynomials on $F$ (often taken as $P_{k-1}(F)$ in classical formulations) (Bohne et al., 5 Jul 2024, Bressan et al., 18 Feb 2025).

Key aspects:

• The space contains all conforming $P_k$ finite element functions as a subspace.
• For $k=1$, in two dimensions, the CR element functions are continuous at edge midpoints and piecewise affine elsewhere.
• Recent work (Bohne et al., 5 Jul 2024) establishes explicit representations of non-conforming basis functions in arbitrary dimension $d \geq 2$ and degree $k\geq 1$, using barycentric coordinates and Jacobi polynomials. Explicitly, the simplex-supported basis for $B^{\text{CR},K}_k$ is:

$$B^{\text{CR},K}_k(x) = \frac{1}{d} \Bigg(-1 + \sum_{z\in \mathcal{V}(K)} P_k^{(0,d-2)}(1-2\lambda_{K,z}(x))\Bigg)$$

on $K$, and zero elsewhere.

• A direct sum decomposition of the CR space into the conforming $P_k$ space and a complementary nonconforming subspace is proved:

$$\text{CR}_k(\mathcal{T}) = S_k(\mathcal{T}) \oplus V_k^{\text{nc}}(\mathcal{T}) \quad \text{for even }k,$$

and an analogous splitting for odd $k$ (with a restriction to functions vanishing at vertices).

2. Degrees of Freedom and Interpolation

Degrees of freedom (DOFs) in the CR setting are constructed to be bidual to the basis functions, typically as integrals of the trial function over facets (edges or faces), weighted by suitable test polynomials of degree up to $k-1$. In two dimensions or for $k=1$, all DOFs can be naturally split into facet and interior simplex integrals, so that the coefficients in the basis expansion corresponding to lower-dimensional facets are fully determined by the corresponding DOFs (Bohne et al., 5 Jul 2024): $J(u) = \frac{2}{|E|} \int_E g_E(x) u(x) dx,$ where $g_E$ is a weight function forming a basis of $P_{k-1}(E)$. This property fails in higher dimensions for $k>1$, as demonstrated via a rigorous nonexistence proof; in those cases, some contributions unavoidably involve integration over the full $d$-simplex (Bohne et al., 5 Jul 2024).

The interpolation operator onto the CR space is constructed locally, mapping functions from $H^1(\Omega)$ or from a discontinuous CR space into the CR space by projection onto the basis via the corresponding DOFs. This operator plays a central role both in the analysis of the approximation properties and in practical implementation.

3. Theoretical Properties and Error Analysis

CR spaces enjoy strong approximation properties. For elliptic PDEs, error estimates in the broken (elementwise) $H^1$ norm are typically optimal, with the convergence order determined by the degree $k$ of the local polynomials (see, for example, (Jr. et al., 2017, Lamichhane, 2014)): $$\|u - u_\text{CR}\|_{H^1_{\text{pw}}(\Omega)} \leq C\, h^r\,\|u\|_{PH^{1+r}(\Omega)},\quad r = \min\{k, s\},$$ where $u$ is the exact solution of the PDE with local Sobolev regularity $s$ (Jr. et al., 2017).

For the surface CR element (nonconforming finite elements on a polyhedral approximation to a manifold), optimal $H^1$- and $L^2$-error estimates are also established (Guo, 2018, Mehlmann, 2023), including rigorous geometric error control: $$\|u^e - u_h\|_h \lesssim h\|f\|_{L^2(\Gamma)},\qquad \|u^e - u_h\|_{L^2(\Gamma_h)} \lesssim h^2\|f\|_{L^2(\Gamma)}.$$

For dual mixed formulations (e.g., the Poisson or Stokes problem), the lowest-order CR space paired with piecewise constants is inf-sup stable and yields quasi-optimal error estimates, both in the (mesh-dependent) natural norm and for the divergence: $$\|(σ-σ_h,\, u-u_h)\|_{nc} \leq C h(\|\sigma\|_{H^2} + |u|_{H^1}),$$ with the local divergence error attaining the best $L^2$-approximation on each element (Barrios et al., 2021).

4. Stability, Discrete Inf-Sup Properties, and High-Order Generalizations

A central property of CR spaces is their stability for saddle-point problems, such as the Stokes system. For triangular meshes in two dimensions, CR spaces of any odd degree $p\geq 3$ satisfy the Ladyzhenskaya–Babuška–Brezzi ("inf-sup") condition, thus forming a stable pair with discontinuous piecewise polynomials of degree $p-1$ for the pressure, regardless of mesh configuration as long as there is at least one interior vertex (Carstensen et al., 2021). In three dimensions, explicit CR bases of polynomial degree $k$ have been constructed and analyzed for stability, with $k=2$ (quadratic velocity) proving to be inf-sup stable while avoiding spurious pressure modes that occur in conforming $(k,k-1)$ elements (Sauter et al., 2022).

For even-degree CR elements, new constructions (imposing orthogonality of jumps with respect to all but one of the degree-$(p-1)$ facet polynomials and including the degree-$p$ polynomial) yield hierarchical, nested bases and permit efficient variable-order and $hp$ adaptive strategies. However, a DG-type stabilization term is required to ensure optimal convergence, with the stabilization parameter considerably smaller than in standard discontinuous Galerkin methods (Bressan et al., 18 Feb 2025).

5. Connections to Raviart–Thomas Spaces and Discrete de Rham Complexes

CR spaces possess deep structural connections to Raviart–Thomas (RT) mixed finite element spaces:

• For the lowest order, the nonconforming gradient of an enriched CR solution coincides exactly with the RT stress approximation (Hu et al., 2014, Ishizaka et al., 2020).
• On the level of algebraic structure, the discrete gradient of CR functions and the divergence of RT functions exhibit precise $L^2$ orthogonality identities, underpinning discrete convex duality and characterizations of projection operators (Bartels et al., 2020).
• These connections are crucial for dual-mixed and pressure-robust methods, as well as for the construction of pressure-robust discretizations with modified CR elements and "lifting operators" mapping CR functions into $H(\text{div};\Omega)$-conforming RT spaces (Ishizaka, 2023).

These relations naturally fit within the framework of discrete de Rham complexes. High-order CR and related discontinuous spaces can be deployed as components in discrete de Rham sequences that retain the correct (co-)homological topology of the underlying continuum problem—crucial for avoiding spurious "harmonic modes" and for devising structure-preserving discretizations in fluid and electromagnetic applications (Perrier, 30 Apr 2024).

6. Practical Applications and Computational Performance

Crouzeix–Raviart spaces have been applied across a range of PDE models:

• Stokes and Navier–Stokes flows: inf-sup stability ensures robust approximation of incompressible fields. Modified anisotropic CR methods yield pressure-independent, anisotropic error estimates and retain optimal convergence even in the presence of large irrotational forces or on non-shape-regular (anisotropic) meshes (Ishizaka, 2023, Ishizaka et al., 2020).
• Plate and shell models: the mixed CR-linear method for Reissner–Mindlin plates eliminates locking without added stabilization (Lamichhane, 2014).
• Phase field–based topology optimization: CR velocity spaces paired with piecewise constant pressure approximations yield efficient discretizations with reduced DOFs and provable convergence/resolution of sharp interfaces, outperforming higher-order conforming methods in computational cost while matching their accuracy (Jin et al., 7 May 2025).
• Eigenvalue problems: With suitable recovery postprocessing, CR and enriched CR discretizations admit asymptotically exact a posteriori error estimators, and corrected eigenvalue approximations achieve superconvergent accuracy (Hu et al., 2019).

The reduced number of DOFs per mesh element, explicit basis constructions, hierarchical and nested properties (with new even-degree variants), and flexibility in coupling with conforming or mixed methods all contribute to the computational efficiency and broad applicability of CR finite element spaces.

7. Extension to Surfaces and High Dimensions

CR elements have been generalized to the discretization of surface PDEs, particularly on triangulated manifolds. The construction of CR spaces on piecewise flat approximations enables optimal discretization of Laplace–Beltrami-type operators and related problems on manifolds, with rigorous geometric error analysis and implementation strategies based on local projections and tangential consistency (Guo, 2018, Mehlmann, 2023). For general polynomial order and arbitrary dimension, recent work has yielded explicit representation of nonconforming basis functions and rigorous analysis of their algebraic properties and decomposition structures (Bohne et al., 5 Jul 2024).

Summary Table: Key Properties of Crouzeix–Raviart Spaces

Aspect	Classical CR ($k=1$)	High-Order/Generalized CR	Surface/Manifold CR
Local Polynomial Degree	1 (Piecewise affine)	$k$ (any degree)	1 (Piecewise affine, surface)
Interelement Continuity	Mean across facets	Orthogonality conditions (jump)	Mean across edge midpoints
Basis Function Type	Facet/edge supported	Simplex- and facet-supported	Edge midpoint continuity
Approximation Space Contains	$P_1$ conforming	$P_k$ conforming	Surface $P_1$ conforming
Inf-sup Stability for Stokes	Yes (2D, $k=1$)	Yes (2D odd $k\geq 3$, 3D $k=2$)	Not directly applicable
Key Applications	Poisson, Stokes, Plates	General elliptic/PDEs, hp-methods	Surface PDEs
Connections to Mixed Spaces	RT equivalence, duality	RT equivalence, deRham complexes	Intrinsic to surface gradients
Basis Representation (2D, $k=1$)	Edges only	Edges and surface integrals	Edge midpoints
Higher-dim Split DOF Possible?	Yes (2D)	No ($d\geq 3$, $k>1$)	Not applicable

In conclusion, the Crouzeix–Raviart finite element spaces, in their various incarnations, offer a mathematically rich and practically effective arena for nonconforming finite element discretizations across a spectrum of PDEs and geometries. The recent advances—explicit high-order bases, nested even-degree spaces, refined interpolation and stabilization, connections to mixed and de Rham complexes, and new manifold and anisotropic generalizations—underscore their foundational role in the finite element landscape.