Curved Hu–Zhang Element in Elasticity

• Curved Hu–Zhang element is a mixed finite element method for elasticity on curved domains, extending the classic Hu–Zhang design with polynomial diffeomorphisms.
• It maps stress spaces from straight triangles to curvilinear boundary elements, ensuring strong symmetry and H(div)-conformity in the discrete formulation.
• A new discrete inf-sup argument in a broken H¹-norm and local boundary-only p-enrichment are introduced to recover optimal convergence rates.

Searching arXiv for recent and foundational papers on the curved Hu–Zhang element and related Hu–Zhang constructions. The curved Hu–Zhang element is a parametric extension of the Hu–Zhang mixed finite element for linear elasticity from straight simplicial meshes to curved domains, designed to preserve strong symmetry of the discrete stress tensor and $H(\mathrm{div})$-conformity on curvilinear triangles. In the recent curved-domain formulation, the stress space is transported from straight triangles to curved boundary elements by polynomial diffeomorphisms, which preserves the defining structural properties of the original Hu–Zhang construction while breaking its purely polynomial character on boundary cells. This loss of polynomial structure changes both the stability analysis and the approximation theory: well-posedness is recovered through a new discrete inf-sup argument in a broken $H^1$-norm, and optimal convergence is obtained for all variables except the stress $L^2$-error unless a local boundary-only $p$-enrichment is introduced (Chen et al., 14 Aug 2025).

1. Definition and problem setting

The Hu–Zhang element is a conforming mixed finite element for the Hellinger–Reissner formulation of linear elasticity, in which the stress is approximated in a symmetric $H(\mathrm{div})$-conforming tensor space and the displacement in a discontinuous vector-valued polynomial space. On a straight triangulation, the global stress space is

$$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$

with bubble space

$$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$

and displacement space

$$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$

(Chen et al., 14 Aug 2025).

For curved domains, the recent construction assumes a bounded domain $\Omega \subset \mathbb{R}^2$ with piecewise $C^{r+1}$ boundary and formulates elasticity in mixed form with

$H^1$0

satisfying

$H^1$1

where

$H^1$2

(Chen et al., 14 Aug 2025).

The central difficulty is geometric. On a curved boundary, replacing the domain by a polygonal approximation introduces a geometric consistency error that may limit the observed convergence order. The curved Hu–Zhang element addresses this by curving the mesh geometry itself: interior triangles remain straight, while boundary triangles are mapped to curvilinear triangles by polynomial diffeomorphisms of degree $H^1$3 (Chen et al., 14 Aug 2025).

2. Straight-triangle Hu–Zhang structure

The curved construction inherits its algebraic and conformity properties from the classical triangular Hu–Zhang element. On straight meshes, a useful characterization is

$H^1$4

which for $H^1$5 is an alternative characterization of the triangular Hu–Zhang element (Aznaran et al., 2024).

Its local unisolvence is expressed through three classes of degrees of freedom: values at vertices, edge moments of normal traction, and interior moments against bubble tensors. On one triangle $H^1$6, the local tensor space $H^1$7 is determined by:

DOF class	Form
Vertex values	$H^1$8 at each vertex
Edge moments	$H^1$9 for $L^2$0
Interior moments	$L^2$1 for $L^2$2

The bubble space satisfies

$L^2$3

and coincides with

$L^2$4

(Aznaran et al., 2024).

The same element admits a template-based reinterpretation in which tensor-valued basis functions are formed as scalar $L^2$5 basis functions multiplied by polytope-associated tensor templates. In two dimensions, the Hu–Zhang space is treated as a symmetric normal-continuous tensor element

$L^2$6

and on the reference triangle $L^2$7 it is assembled from vertex, edge, and cell contributions: $L^2$8 (Sky et al., 2024).

A structurally important feature is that the vertex basis is fully continuous, while edge basis functions enforce continuity of the normal trace and allow the edge tangential-tangential component to jump. The template paper explicitly notes that, for the Hu–Zhang element on triangles, the vertex basis must be fully continuous because a conforming $L^2$9-type symmetric tensor space on simplices cannot be minimally regular (Sky et al., 2024).

3. Curved-domain construction

The curved Hu–Zhang element extends this straight-triangle space to curvilinear boundary cells through elementwise geometric maps. The construction assumes a shape-regular triangulation in which each boundary triangle has at most one boundary edge: Hypothesis 3.1. Each boundary triangle has at most two boundary vertices, hence at most one edge on $p$0 (Chen et al., 14 Aug 2025).

For each straight triangle $p$1, a degree-$p$2 diffeomorphism

$p$3

is introduced so that:

• on interior elements, $p$4,
• on interior edges of boundary elements, $p$5 is also the identity,
• on the boundary edge, it interpolates a parametrization of $p$6

(Chen et al., 14 Aug 2025).

The curved mesh is therefore

$p$7

To compare different geometry orders, the elementwise map

$p$8

is used, together with the estimate

$p$9

and the analogous estimate for the inverse (Chen et al., 14 Aug 2025).

For $H(\mathrm{div})$0, the curved stress space is defined by pullback to the straight element: $H(\mathrm{div})$1 with

$H(\mathrm{div})$2

The curved displacement space is

$H(\mathrm{div})$3

(Chen et al., 14 Aug 2025).

This construction preserves the two defining structural properties of the original method. Symmetry is retained because the stress tensor is represented as symmetric on the reference element and transported in a way that keeps symmetry. $H(\mathrm{div})$4-conformity is preserved because the bubble space on a curved boundary element still has vanishing normal component on the interior edges where the geometric map is the identity. In particular, for a boundary element $H(\mathrm{div})$5 and its interior edge $H(\mathrm{div})$6,

$H(\mathrm{div})$7

(Chen et al., 14 Aug 2025).

4. Mapping theory and the role of non-affine transformations

The curved Hu–Zhang element is closely related to a broader question: how symmetric $H(\mathrm{div})$8-conforming tensor elements should be mapped from reference simplices to physical simplices when the geometry is not affine. The template framework of Kirby, Mitchell, and coauthors identifies the Hu–Zhang element as a case where the standard double contravariant Piola is insufficient: $H(\mathrm{div})$9 That mapping preserves the normal-normal trace, but Hu–Zhang basis functions include fully $$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$0 vertex functions built from a Cartesian symmetric basis as well as edge functions that distinguish tangential, normal, and mixed components. On curved or non-affine elements, a single universal double Piola map does not preserve these mixed continuity properties consistently across all polytope classes (Sky et al., 2024).

The alternative proposed in the template framework is an edge-wise transformation

$$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$1

where $$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$2 are the reference edge tangent and normal and $$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$3 the physical edge tangent and normal. This mapping satisfies

$$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$4

hence it preserves symmetry and transports the edge-based trace structure correctly (Sky et al., 2024).

That paper explicitly states that the polytopal template construction enables a transformation from a reference simplex to a non-affine simplex in the physical mesh, which is not available through the standard double Piola map alone. It also warns that “for non-affine case, it is necessary to choose a hierarchical basis” because different polytope classes may be transformed differently, and a non-hierarchical basis could lose the partition-of-unity or constant-space property (Sky et al., 2024).

This suggests a conceptual distinction between two related developments. The template paper supplies a template-aware transport mechanism for Hu–Zhang-type bases on non-affine simplices (Sky et al., 2024), whereas the later curved-domain elasticity paper provides a full mixed finite element theory on curved domains, including stability and convergence analysis for the resulting non-polynomial spaces (Chen et al., 14 Aug 2025).

5. Stability, exact-sequence structure, and the new inf-sup theory

On straight triangulations, the Hu–Zhang element has a strong exact-sequence interpretation. In two dimensions, for mixed boundary conditions, the discrete elasticity complex can be written as

$$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$5

where $$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$6 is the Hu–Zhang stress space (Aznaran et al., 2024). The same work proves the main $$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$7-stability statement: $$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$8 with $$\Sigma_h = \left\{ \sigma\in H(\mathrm{div},\Omega;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\ \sigma_c\in H^1(\Omega;\mathbb{S}),\ \sigma_c|_K\in P_k(K;\mathbb{S}),\ \sigma_b|_K\in \Sigma_{k,b}(K) \right\},$$9 independent of $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$0 and $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$1 (Aznaran et al., 2024).

That result depends on a polynomial framework unavailable on curved boundary elements. The curved-domain paper therefore replaces the classical argument with a new one. The main obstacle is that the curved spaces are no longer polynomial on boundary elements, so the classical Hu–Zhang proof of stability does not apply directly. The replacement is a new discrete inf-sup theorem in a broken $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$2-norm: for every $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$3, there exists $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$4 such that

$$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$5

(Chen et al., 14 Aug 2025).

This is the key analytical step because the perturbation between the curved bilinear form $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$6 and the straight bilinear form $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$7 is naturally controlled in the broken $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$8-norm rather than the full $$\Sigma_{k,b}(K):=\{\tau\in P_k(K;\mathbb{S}):\ \tau\nu|_{\partial K}=0\},$$9-norm. For sufficiently small $$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$0 and geometry order $$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$1, one then obtains

$$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$2

which yields the discrete inf-sup condition and hence well-posedness together with coercivity of $$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$3 (Chen et al., 14 Aug 2025).

A common misconception is that the curved element is simply the straight Hu–Zhang space composed with an isoparametric map and therefore inherits the affine theory unchanged. The recent analysis shows that this is not the case: the non-polynomial structure on curved boundary elements is the source of both the new inf-sup proof and the modified error behavior (Chen et al., 14 Aug 2025).

6. Approximation properties, suboptimality, and enrichment

On affine meshes, the classical Hu–Zhang method satisfies the standard optimal estimate

$$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$4

(Chen et al., 14 Aug 2025). On curved meshes, the behavior changes because

$$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$5

This failure of exact divergence compatibility is the reason the stress $$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$6-error becomes suboptimal (Chen et al., 14 Aug 2025).

Under suitable regularity assumptions, the curved-domain paper proves an energy-type estimate with

$$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$7

namely

$$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$8

It also proves a mesh-dependent superclose estimate

$$V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}$$9

If $\Omega \subset \mathbb{R}^2$0 is convex, then

$\Omega \subset \mathbb{R}^2$1

and for a postprocessed displacement $\Omega \subset \mathbb{R}^2$2,

$\Omega \subset \mathbb{R}^2$3

(Chen et al., 14 Aug 2025).

The stress $\Omega \subset \mathbb{R}^2$4-error exhibits a half-order loss: $\Omega \subset \mathbb{R}^2$5 rather than the affine optimal rate $\Omega \subset \mathbb{R}^2$6 (Chen et al., 14 Aug 2025). The paper attributes this directly to the incompatibility $\Omega \subset \mathbb{R}^2$7.

To repair the loss, the authors propose local polynomial enrichment on boundary elements only: interior elements retain degree $\Omega \subset \mathbb{R}^2$8, while curved boundary elements use degree $\Omega \subset \mathbb{R}^2$9 in both stress and displacement spaces. The enriched spaces $C^{r+1}$0 and $C^{r+1}$1 then recover the optimal stress rate

$C^{r+1}$2

with only a small additional cost because the enrichment is confined to the boundary layer (Chen et al., 14 Aug 2025).

The numerical experiments on the unit disk and a nonconvex three-leaf domain confirm the theory: the standard curved Hu–Zhang element exhibits the predicted rate $C^{r+1}$3 in the stress $C^{r+1}$4-error, while the enriched boundary-element version recovers the missing half-order (Chen et al., 14 Aug 2025).

7. Related developments and broader significance

Several adjacent results clarify why the curved Hu–Zhang element takes its present form. The 2018 work on partial relaxation of $C^{r+1}$5 vertex continuity shows that the original triangular Hu–Zhang element is “too continuous at vertices” for nestedness under newest-vertex bisection. The remedy is to relax only the tangential-tangential vertex continuity at newly created midpoint vertices while preserving continuity of the normal-related components

$C^{r+1}$6

and allowing

$C^{r+1}$7

to split into one-sided basis functions (Hu et al., 2018). This result concerns adaptive straight triangulations rather than curved geometry, but it highlights a defining theme of Hu–Zhang analysis: conformity is tied to a delicate separation between normal and tangential stress components.

The 2024 $C^{r+1}$8-stability paper adds a complementary perspective by proving uniformly $C^{r+1}$9-stable inf-sup bounds for the two-dimensional Hu–Zhang pair on affine triangular meshes and constructing $H^1$00-bounded commuting cochain projections and $H^1$01-stable Hodge decompositions within the BGG framework (Aznaran et al., 2024). That theory demonstrates that the straight Hu–Zhang element has a robust exact-sequence structure independent of both $H^1$02 and $H^1$03, which helps explain why curvature-induced loss of polynomial compatibility becomes the dominant new issue on curved domains.

A further related development, though not a Hu–Zhang construction itself, is the curved-edge Virtual Element framework of Mascotto and collaborators. That work emphasizes curved-edge traces defined as restrictions of planar polynomials, generator-based representations rather than strict nodal DOFs, and careful stabilization on curved interfaces (Veiga et al., 2019). A plausible implication is that these principles are structurally aligned with future tensor-valued curved $H^1$04-conforming designs, especially where curved traces and interface regularity matter.

In this broader context, the curved Hu–Zhang element occupies a specific position: it is not merely a geometric variant of a known affine element, but a mixed finite element method in which curving the mesh preserves geometry while breaking exact polynomial divergence compatibility. The resulting method remains strongly symmetric, $H^1$05-conforming, and stable on curved domains, but requires new analysis and, for fully optimal stress $H^1$06-accuracy, a targeted boundary-only $H^1$07-enrichment (Chen et al., 14 Aug 2025).