---
title: Curved Hu–Zhang Element in Elasticity
url: https://www.emergentmind.com/topics/curved-hu-zhang-element
type: topic
---

# Curved Hu–Zhang Element in Elasticity

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 [2508.10674].

## 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
\[
V_h:=\{v\in L^2(\Omega;\mathbb{R}^2):\ v|_K\in P_{k-1}(K;\mathbb{R}^2)\}
\]
[2508.10674].

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
\[
(\sigma,u)\in \Sigma\times V := H(\mathrm{div},\Omega;\mathbb{S})\times L^2(\Omega;\mathbb{R}^2),
\]
satisfying
\[
\begin{cases}
a(\sigma,\tau)+b(\tau,u)=0, & \forall \tau\in \Sigma,\\[2mm]
b(\sigma,v)=-(f,v), & \forall v\in V,
\end{cases}
\]
where
\[
a(\sigma,\tau) := (A\sigma,\tau),\qquad b(\tau,v):=(\mathrm{div}\,\tau,v)
\]
[2508.10674].

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 \(m\) [2508.10674].

## 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
\[
\Sigma^p := \{ \sigma \in H(\mathrm{div};\mathbb{S}) : \sigma|_K \in \mathcal{P}_p(K;\mathbb{S})\ \forall K,\ \text{$\sigma$ is \(C^0\) at element vertices}\},
\]
which for \(p\ge 3\) is an alternative characterization of the triangular Hu–Zhang element [2409.17414].

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 \(K\), the local tensor space \(\mathcal{P}_p(K;\mathbb{S})\) is determined by:

| DOF class | Form |
|---|---|
| Vertex values | \(\sigma(x)\) at each vertex |
| Edge moments | \(\int_\gamma (\sigma n)\cdot r\,ds\) for \(r\in \mathcal{P}_{p-2}(\gamma;\mathbb{V})\) |
| Interior moments | \(\int_K \sigma:\tau\,dx\) for \(\tau \in {}_0^p(K;\mathbb{S})\) |

The bubble space satisfies
\[
{}_0^p(K;\mathbb{S}) := \mathcal{P}_p(K;\mathbb{S}) \cap \{ \tau : \tau n|_{\partial K}=0\},
\qquad
\dim {}_0^p(K;\mathbb{S}) = \frac{3}{2}p(p-1),
\]
and coincides with
\[
\lambda_2\lambda_3\mathcal{P}_{p-2}(K)(t_1\otimes t_1) \oplus \lambda_1\lambda_3\mathcal{P}_{p-2}(K)(t_2\otimes t_2) \oplus \lambda_1\lambda_2\mathcal{P}_{p-2}(K)(t_3\otimes t_3)
\]
[2409.17414].

The same element admits a template-based reinterpretation in which tensor-valued basis functions are formed as **scalar \(C^0\) basis functions multiplied by polytope-associated tensor templates**. In two dimensions, the Hu–Zhang space is treated as a **symmetric normal-continuous tensor element**
\[
HZ^p(A) \subset H\bigl(\mathrm{div};A,\mathrm{Sym}(2)\bigr),
\]
and on the reference triangle \(\Gamma\) it is assembled from vertex, edge, and cell contributions:
\[
HZ^p(\Gamma) = \left\{ \bigoplus_{i=0}^2 V_i^p(\Gamma)\otimes T_i^{\,\mathrm{sym},\nu} \right\} \oplus \left\{ \bigoplus_{j\in\mathcal J} E_j^p(\Gamma)\otimes T_j^{\,\mathrm{sym},\nu} \right\} \oplus C_{012}^p(\Gamma)\otimes T_{012}^{\,\mathrm{sym},\nu}
\]
[2405.10402].

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 \(H(\mathrm{div})\)-type symmetric tensor space on simplices cannot be minimally regular [2405.10402].

## 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 \(\partial\Omega\) [2508.10674].

For each straight triangle \(K^1\), a degree-\(m\) diffeomorphism
\[
F_K^m:K^1\to K^m
\]
is introduced so that:

- on interior elements, \(F_K^m=\mathrm{id}\),
- on interior edges of boundary elements, \(F_K^m\) is also the identity,
- on the boundary edge, it interpolates a parametrization of \(\partial\Omega\)

[2508.10674].

The curved mesh is therefore
\[
\mathcal{T}^m=\{K^m=F_K^m(K^1):K^1\in\mathcal{T}\}.
\]
To compare different geometry orders, the elementwise map
\[
\Phi^{lm} : \Omega^l\to\Omega^m,\qquad \Phi^{lm}|_{K^l}=F_K^m\circ (F_K^l)^{-1}
\]
is used, together with the estimate
\[
\|\nabla^s(\Phi^{lm}-\mathrm{id})\|_{L^\infty(K^l)}\lesssim h^{l+1-s}, \qquad 0\le s\le l+1
\]
and the analogous estimate for the inverse [2508.10674].

For \(k\ge 3\), the curved stress space is defined by pullback to the straight element:
\[
\Sigma_h^m = \left\{ \sigma\in L^2(\Omega^m;\mathbb{S}) : \sigma=\sigma_c+\sigma_b,\  \sigma_c\in H^1(\Omega^m;\mathbb{S}),\ \sigma_c|_{K^m}\circ F_K^m\in P_k(K^1;\mathbb{S}),\ \sigma_b|_{K^m}\in \Sigma_{k,b}(K^m) \right\},
\]
with
\[
\Sigma_{k,b}(K^m):= \{\tau\in L^2(K^m;\mathbb{S}) : \tau\circ F_K^m\in \Sigma_{k,b}(K^1)\}.
\]
The curved displacement space is
\[
V_h^m:=\{v\in L^2(\Omega^m;\mathbb{R}^2):\ v\circ F^m\in V_h^1\}
\]
[2508.10674].

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. \(H(\mathrm{div})\)-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 \(K^m\) and its interior edge \(E\),
\[
F_K^m|_E=\mathrm{id} \quad\Longrightarrow\quad \tau\cdot\nu|_E=0 \quad \forall \tau\in \Sigma_{k,b}(K^m)
\]
[2508.10674].

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

The curved Hu–Zhang element is closely related to a broader question: how symmetric \(H(\mathrm{div})\)-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:
\[
Y = \frac{1}{(\det J)^2} J\,\Upsilon\,J^T.
\]
That mapping preserves the normal-normal trace, but Hu–Zhang basis functions include fully \(C^0\) 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 [2405.10402].

The alternative proposed in the template framework is an **edge-wise transformation**
\[
Y = T_j\,\Upsilon\,T_j^T, \qquad
T_j = \frac{1}{\tau^2}\,(t\otimes \tau + n\otimes \nu), \qquad j\in\mathcal J,
\]
where \(\tau,\nu\) are the reference edge tangent and normal and \(t,n\) the physical edge tangent and normal. This mapping satisfies
\[
T(\tau\otimes\tau)T^T=t\otimes t, \qquad
T\sym(\tau\otimes\nu)T^T=\sym(t\otimes n), \qquad
T(\nu\otimes\nu)T^T=n\otimes n,
\]
hence it preserves symmetry and transports the edge-based trace structure correctly [2405.10402].

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 [2405.10402].

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 [2405.10402], 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 [2508.10674].

## 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
\[
0 \longrightarrow \mathcal{P}_{1,\Gamma}(\Omega) \longrightarrow Q_\Gamma^{p+2} \stackrel{\airy}{\longrightarrow} \Sigma_\Gamma^p \stackrel{\mathrm{div}{\longrightarrow} V_\Gamma^{p-1} \longrightarrow 0,
\]
where \(\Sigma_\Gamma^p\) is the Hu–Zhang stress space [2409.17414]. The same work proves the main \(hp\)-stability statement:
\[
\forall u\in V_\Gamma^{p-1},\ \exists \sigma\in \Sigma_\Gamma^p \text{ such that } \mathrm{div}\,\sigma=u,\qquad \|\sigma\|_{\mathrm{div},\Omega}\le C\|u\|_{0,\Omega},
\]
with \(C\) independent of \(h\) and \(p\) [2409.17414].

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 \(H^1\)-norm**: for every \(v_h\in V_h^1\), there exists \(\tau\in\Sigma_h^1\) such that
\[
\mathrm{div}\,\tau=v_h,\qquad \|\tau\|_{1,\mathcal{T}}\lesssim \|v_h\|_0
\]
[2508.10674].

This is the key analytical step because the perturbation between the curved bilinear form \(b^m\) and the straight bilinear form \(b^1\) is naturally controlled in the broken \(H^1\)-norm rather than the full \(H(\mathrm{div})\)-norm. For sufficiently small \(h\) and geometry order \(m>1\), one then obtains
\[
b^m(\tau,v_h)\gtrsim \|v_h\|_0^2, \qquad \|\tau\|_{1,\mathcal{T}}\lesssim \|v_h\|_0,
\]
which yields the discrete inf-sup condition and hence well-posedness together with coercivity of \(a^m\) [2508.10674].

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 [2508.10674].

## 6. Approximation properties, suboptimality, and enrichment

On affine meshes, the classical Hu–Zhang method satisfies the standard optimal estimate
\[
\|\sigma-\sigma_h\|_{0} + h\|\sigma-\sigma_h\|_{H(\mathrm{div})} + h\|u-u_h\|_0 \lesssim h^{k+1}
\]
[2508.10674]. On curved meshes, the behavior changes because
\[
\mathrm{div}\,\Sigma_h^m \not\subset V_h^m.
\]
This failure of exact divergence compatibility is the reason the stress \(L^2\)-error becomes suboptimal [2508.10674].

Under suitable regularity assumptions, the curved-domain paper proves an energy-type estimate with
\[
q=\min\{k,m+\tfrac12,s-1\},
\]
namely
\[
\|\sigma-\sigma_h\|_{H(\mathrm{div})}+\|u-u_h\|_0 \lesssim h^q\|u\|_{H^{q+2}(\Omega)}.
\]
It also proves a mesh-dependent superclose estimate
\[
\|\sigma-\sigma_h\|_{0,h,m} + |Q_hu-u_h|_{1,h,m} \lesssim h^{\min\{k+\frac12,m+\frac12,s\}}\|u\|_{H^{s+1}(\Omega)}.
\]
If \(\Omega\) is convex, then
\[
\|Q_hu-u_h\|_0 \lesssim h^{\min\{k+\frac32,m+1,s+1\}},
\qquad
\|u-u_h\|_0 \lesssim h^{\min\{k,m+1,s+1\}},
\]
and for a postprocessed displacement \(u_h^{*}\),
\[
\|u-u_h^{*}\|_0 \lesssim h^{\min\{k+\frac32,m+1,s+1\}}
\]
[2508.10674].

The stress \(L^2\)-error exhibits a half-order loss:
\[
\|\sigma-\sigma_h\|_{0} = \mathcal{O}\!\left(h^{k+\frac12}\right)
\]
rather than the affine optimal rate \(h^{k+1}\) [2508.10674]. The paper attributes this directly to the incompatibility \(\mathrm{div}\,\Sigma_h^m \not\subset V_h^m\).

To repair the loss, the authors propose **local polynomial enrichment on boundary elements only**: interior elements retain degree \(k\), while curved boundary elements use degree \(k+1\) in both stress and displacement spaces. The enriched spaces \(\widetilde\Sigma_h^m\) and \(\widetilde V_h^m\) then recover the optimal stress rate
\[
\|\sigma-\sigma_h\|_0 = \mathcal{O}(h^{k+1})
\]
with only a small additional cost because the enrichment is confined to the boundary layer [2508.10674].

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 \(\min\{k+\tfrac12,m+\tfrac12\}\) in the stress \(L^2\)-error, while the enriched boundary-element version recovers the missing half-order [2508.10674].

## 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 \(C^0\) 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
\[
n_e^T \tau n_e,\qquad t_e^T\tau n_e,
\]
and allowing
\[
t_e^T\tau t_e
\]
to split into one-sided basis functions [1807.08090]. 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 \(hp\)-stability paper adds a complementary perspective by proving **uniformly \(hp\)-stable inf-sup bounds** for the two-dimensional Hu–Zhang pair on affine triangular meshes and constructing \(hp\)-bounded commuting cochain projections and \(hp\)-stable Hodge decompositions within the BGG framework [2409.17414]. That theory demonstrates that the straight Hu–Zhang element has a robust exact-sequence structure independent of both \(h\) and \(p\), 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 [1910.10184]. A plausible implication is that these principles are structurally aligned with future tensor-valued curved \(H(\mathrm{div})\)-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, \(H(\mathrm{div})\)-conforming, and stable on curved domains, but requires new analysis and, for fully optimal stress \(L^2\)-accuracy, a targeted boundary-only \(p\)-enrichment [2508.10674].

Source: https://www.emergentmind.com/topics/curved-hu-zhang-element