arXiv subjects

Xuehai Huang

Publications and source records attributed to Xuehai Huang.

At least 19 recordsLinked to original sources

arXiv2026

An interior penalty nonconforming finite element method for strain gradient elasticity

An interior penalty nonconforming finite element method is developed for a strain gradient elasticity (SGE) model in arbitrary dimensions, using an $H^1$-nonconforming displacement element built from vector-valued quadratic polynomials enriched by divergence-free face bubbles. Rigorous analysis establishes optimal and robust error estimates with respect to both the Lamé coefficient $λ$ and the size parameter $ι$. The underlying divergence-commuting structure further leads to nonconforming finite element Stokes complexes in two and three dimensions. Numerical experiments support the predicted convergence behavior and parameter robustness, while a circular-hole benchmark under uniaxial tension illustrates the applicability of the method to a multiply connected domain.

math.NA↗

arXiv2026

Weakly Symmetric and Traceless Tangential-Normal Tensor Finite Elements: Application to the Brinkman Equations

We develop a family of weakly symmetric and pointwise traceless tangential-normal tensor finite elements in arbitrary space dimension and for all polynomial orders. Symmetry is imposed through local cell moments, while the only globally coupled stress degrees of freedom are tangential-normal facet moments; no vertex degrees of freedom are required. In dimensions three and higher, a lowest-order linear enrichment restores the rigid-motion facet control required for discrete Korn stability. As a principal application, we construct a distributional mixed method for the incompressible Brinkman equations using the physical viscous stress. Coupled with divergence-conforming BDM velocities and discontinuous pressures, the method is stabilization-free, uniformly stable with respect to the viscosity parameter, exactly divergence-free, and pressure-robust. We establish optimal-order error estimates in the natural norms. Under suitable parameter-explicit regularity assumptions, we also obtain a parameter-uniform boundary-layer estimate with optimal Darcy approximation order. Relaxing tangential-normal continuity yields an algebraically equivalent stress-hybridized formulation and a stabilization-free virtual element realization.

math.NA↗

arXiv2026

Tangential-Normal Decompositions of the Second Family of Finite Element Differential Forms

This paper introduces a novel tangential-normal ($t$-$n$) decomposition for the second family of finite element differential forms, presenting a new framework for constructing bases in finite element exterior calculus. The main contribution is the development of a $t$-$n$ basis in which degrees of freedom and shape functions are explicitly dual, a property that streamlines stiffness matrix assembly and enhances the efficiency of interpolation and numerical integration. Additionally, the integration of the well-documented Lagrange element basis supports practical implementation of finite element differential forms in applications.

math.NA↗

arXiv2026

Superconvergent and Divergence-Free Mixed Finite Element Methods for The Stokes Equation

This paper develops divergence-free mixed finite element methods for the Stokes equation. Using H(div)-conforming velocities and discontinuous pressures ensures the inf-sup condition for the velocity--pressure pair and yields pointwise divergence-free velocities. However, this choice makes the vector Laplacian difficult to discretize. Inspired by mass-conserving mixed formulations with stresses, tangential--normal continuous traceless tensor elements are used to discretize the vector Laplacian. An inf-sup condition for the weak div operator between the stress and velocity spaces is then proved. Two key properties characterize the scheme. First, the stress--velocity inf-sup stability gives a stable discretization of the vector Laplacian without additional stabilization, unlike discontinuous Galerkin or virtual element methods. Second, the scheme has the property that if a stress field is distributionally divergence-free against the discrete divergence-free velocity space, then it is also distributionally divergence-free against the continuous divergence-free velocity space. This property decouples the stress and velocity errors and leads to superconvergence. As a result, optimal-order error estimates are obtained for the stress, while the discrete velocity is superclose to its H(div) interpolant. The projected-pressure error estimate is optimal with Raviart-Thomas velocities and superconvergent with Brezzi-Douglas-Marini velocities, while local postprocessing yields an elementwise divergence-free velocity with higher-order convergence. Numerical experiments confirm the theoretical results.

math.NA↗

arXiv2026

A Lowest-Order Robust Mixed Finite Element Method with a Third-Order Tensor Variable for Strain Gradient Elasticity

A lowest-order mixed finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimensions. We take the physically meaningful third-order double stress tensor $\boldsymbolΦ:=ι^2\operatorname{grad}\boldsymbolσ(\boldsymbol{u})\in\mathbb{S}\otimes\mathbb{R}^d$ as a primary variable and derive a distributional mixed formulation. The double stress is approximated by an $\mathbb{S}\otimes\mathbb{R}^d$-valued extension of the lowest-order Raviart--Thomas element, while the displacement is approximated by the vector-valued linear Crouzeix--Raviart element. Thus, the method avoids both high-degree bubble enrichment and a Nitsche-type treatment of the higher-order boundary condition. We establish parameter-robust discrete stability, an optimal first-order error estimate for fixed parameters, and a complementary parameter-uniform $\mathcal{O}(ι^{1/2}+h)$ error estimate with constants independent of both the size parameter $ι$ and the Lamé coefficient $λ$. In the boundary-layer regime $ι^{1/2}\lesssim h$, the latter retains a first-order convergence rate in $h$. We also develop a local quadratic post-processing and a hybridized formulation. Numerical experiments in two and three dimensions support the theoretical results.

math.NA↗

arXiv2026

Symmetric-Tensor Distributional Mixed Method for Fourth-Order Elliptic Singular Perturbation Problem

A symmetric-tensor distributional mixed method for a fourth-order elliptic singular perturbation problem is developed in this paper. The moment variable is approximated by normal-normal continuous symmetric tensor elements, while the scalar variable is represented by an H^1-nonconforming virtual element space coupled with a polynomial multiplier on interior codimension-two subsimplices. Optimal parameter-uniform error estimates are derived, independent of the presence of boundary layers. A hybridized form is further shown to be equivalent to stabilization-free weak Galerkin and H^2-nonconforming virtual element formulations. In two dimensions, we establish a close connection between the distributional mixed method and the classical Hellan-Herrmann-Johnson (HHJ) method by identifying the scalar virtual element-multiplier pair with the Lagrange finite element space. Consequently, the proposed method extends the two-dimensional HHJ framework to any spatial dimension d >= 2. Three-dimensional numerical experiments support the theoretical convergence and robustness estimates. A two-dimensional adaptive constant-load benchmark on an L-shaped polygonal domain tests the method on a non-manufactured nonsmooth problem and shows mesh concentration near the reentrant corner and, for small epsilon, boundary refinement at the expected O(epsilon) scale.

math.NA↗

arXiv2026

Low-regularity finite element elasticity complexes with hybridizable stresses on tetrahedral Alfeld splits

Finite element elasticity complexes of low regularity are constructed on tetrahedral Alfeld splits. In comparison with existing three-dimensional elasticity complexes on such splits, the complexes constructed here lower both the Sobolev regularity and the polynomial degrees, while ending in a hybridizable $H({\rm div};\mathbb S)$-conforming symmetric stress space with no vertex degrees of freedom. The construction is obtained from local Bernstein-Gelfand-Gelfand arguments applied to polynomial de Rham complexes on the Alfeld split. Two local polynomial elasticity complexes are proved: an $H^2$-$H^1({\rm inc})$ complex and a lower-regularity $H^1({\rm curl})$-$H({\rm inc}^+)$ complex. Their bubble subcomplexes and dimension formulas are derived. These local exact sequences lead to unisolvent finite elements for the displacement and incompatibility spaces and to global finite element subcomplexes of the corresponding elasticity sequences. In the lowest-order $H^1({\rm curl})$-$H({\rm inc}^+)$ finite element complex, the $H({\rm inc}^+;\mathbb S)$-conforming tensor space is piecewise cubic. At the same order, the terminal stress-displacement pair recovers the Johnson-Mercier-Křížek element, while the construction covers higher-order hybridizable symmetric stresses for all $k\ge1$. A second family gives a low-regularity $H^1$-$H({\rm inc})$ finite element complex for the standard elasticity sequence for all $k\ge2$. Commuting interpolation diagrams are established for both global complexes.

math.NA↗

arXiv2026

Hybridizable Staggered Discontinuous Galerkin Methods for Polyharmonic Equations on Polytopes

Hybridizable staggered discontinuous Galerkin methods are developed for arbitrary-order polyharmonic equations $(-Δ)^m u=f$ on shape-regular polytopal meshes in $\mathbb R^d$, for any $m\ge1$, $d\ge2$, and polynomial degree $k\ge0$. The method uses the mixed variable $σ=\nabla^m u$ and a staggered primal--dual mesh to impose complementary continuity on scalar and tensor unknowns, without restrictions such as $d\ge m$. Local trace and bubble enrichments stabilize low-order tensor spaces without adding global unknowns. Hybridization localizes the tensor variable and yields an equivalent stabilization-free weak Galerkin formulation. Well-posedness and optimal energy error estimates are proved, and numerical experiments on polygonal and tetrahedral meshes confirm the predicted rates.

math.NA↗

arXiv2026

Nonconforming Linear Element Method for a Generalized Tensor-Valued Stokes Equation with Application to the Triharmonic Equation

A nonconforming linear element method is developed for a three-dimensional generalized tensor-valued Stokes equation associated with the Hessian complex in this paper. A discrete Helmholtz decomposition for the piecewise constant space of traceless tensors is established, ensuring the well-posedness of the nonconforming method, and optimal error estimates are derived. Building on this, a low-order decoupled finite element method for the three-dimensional triharmonic equation is constructed by combining the Morley-Wang-Xu element methods for the biharmonic subproblems with the proposed nonconforming linear element method. Numerical experiments confirm the theoretical convergence rates.

math.NA↗

arXiv2026

An Optimal and Robust Nonconforming Finite Element Method for the Strain Gradient Elasticity

An optimal and robust low-order nonconforming finite element method is developed for the strain gradient elasticity (SGE) model in arbitrary dimensions. An $H^2$-nonconforming quadratic vector-valued finite element in arbitrary dimensions is constructed, which together with the Nitsche's technique, is applied for solving the SGE model. The resulting nonconforming finite element method is optimal and robust with respect to the Lamé coefficient $λ$ and the size parameter $ι$, as confirmed by numerical results. Additionally, nonconforming finite element discretization of the smooth Stokes complex in two and three dimensions is devised.

math.NA↗

arXiv2026

A Staggered Discontinuous Galerkin Method for linear elasticity problem on Polytopal Meshes

This paper develops a novel staggered discontinuous Galerkin (SDG) method for linear elasticity based on the Hellinger-Reissner variational principle. We construct symmetric stress spaces with normal continuity across element boundaries on arbitrary polytopal meshes, while approximating the displacement field using piecewise polynomial functions defined on the same meshes. The method is locking-free and satisfies a local balance of linear momentum and angular momentum. We present a comprehensive theoretical analysis, including proofs of stability and error estimates. The formulation admits a hybridizable structure, which significantly simplifies the numerical implementation. Numerical experiments validate the theoretical results and demonstrate the effectiveness of the proposed approach.

math.NA↗

arXiv2026

Explicit Planar Finite Element Elasticity Complexes and $C^1$ Elements on Barycentric Refinements

The exact-sequence structure behind the Arnold--Douglas--Gupta family of higher-order mixed finite elements for plane elasticity on barycentric refinements is made explicit. On each macro triangle, the symmetric stress space is obtained by enriching polynomial stresses with three locally supported functions. We derive closed-form formulas for these enrichments and identify explicit Airy potentials that generate them. This leads to a concrete Hsieh--Clough--Tocher type $C^1$ potential space whose Airy image is exactly the Arnold--Douglas--Gupta stress space. By enforcing single-valued degrees of freedom, we obtain global spaces and a fully explicit finite element elasticity complex on simply connected domains. As a consequence, we construct a new family of $C^1$ finite elements on barycentric refinements, including quadratic, cubic, quartic, and higher-order elements.

math.NA↗

arXiv2025

Low-order finite element complex with application to a fourth-order elliptic singular perturbation problem

A low-order nonconforming finite element discretization of a smooth de Rham complex starting from the $H^2$ space in three dimensions is proposed, involving an $H^2$-nonconforming finite element space, a new tangentially continuous $H^1$-nonconforming vector-valued finite element space, the lowest-order Raviart-Thomas space, and piecewise constant functions. While nonconforming for the smooth complex, the discretization conforms to the classical de Rham complex. It is applied to develop a decoupled mixed finite element method for a fourth-order elliptic singular perturbation problem, focusing on the discretization of a generalized singularly perturbed Stokes-type equation. In contrast to Nitsche's method, which requires additional stabilization to handle boundary layers, the nodal interpolation operator for the lowest-order Nédélec element of the second kind is introduced into the discrete bilinear forms. This modification yields a decoupled mixed method that achieves optimal convergence rates uniformly with respect to the perturbation parameter, even in the presence of strong boundary layers, without requiring any additional stabilization.

math.NA↗

arXiv2025

Hybridizable Symmetric Stress Elements on the Barycentric Refinement in Arbitrary Dimensions

Hybridizable $H(\textrm{div})$-conforming finite elements for symmetric tensors on simplices with barycentric refinement are developed in this work for arbitrary dimensions and any polynomial order. By employing barycentric refinement and an intrinsic tangential-normal ($t$-$n$) decomposition, novel basis functions are constructed to redistribute degrees of freedom while preserving $H(\textrm{div})$-conformity and symmetry, and ensuring inf-sup stability. These hybridizable elements enhance computational flexibility and efficiency, with applications to mixed finite element methods for linear elasticity.

math.NA↗

arXiv2025

Robust and Optimal Mixed Methods for a Fourth-Order Elliptic Singular Perturbation Problem

A series of robust and optimal mixed methods based on two mixed formulations of the fourth-order elliptic singular perturbation problem are developed in this paper. First, a mixed method based on a second-order system is proposed without relying on Nitsche's technique or interpolations. Robust and optimal error estimates are derived using an $L^2$-bounded interpolation operator for tensors. Then, its connections to other discrete methods, including weak Galerkin methods and a mixed finite element method based on a first-order system, are established. Finally, numerical experiments are provided to validate the theoretical results.

math.NA↗

arXiv2025

Finite element conformal complexes in three dimensions

This paper extends the Bernstein-Gelfand-Gelfand (BGG) framework to the construction of finite element conformal Hessian complexes and conformal elasticity complexes in three dimensions involving conformal tensors (i.e., symmetric and traceless tensors). These complexes incorporate higher-order differential operators, including the linearized Cotton-York operator, and require conformal tensor spaces with nontrivial smoothness and trace conditions. A novel application of the discrete BGG framework, combined with the geometric decomposition of bubble spaces and a reduction operation, to local bubble finite element complexes is introduced. This yields simpler and more tractable constructions than global BGG-based approaches, and leads to the bubble conformal complexes. Building on these bubble conformal complexes and the associated face bubble complexes, finite element conformal Hessian complexes and conformal elasticity complexes with varying degrees of smoothness are systematically developed. The resulting complexes support stable and structure-preserving numerical methods for applications in relativity, Cosserat elasticity, and fluid mechanics.

math.NA↗

arXiv2025

Implementation and Basis Construction for Smooth Finite Element Spaces

The construction of $C^m$ conforming finite elements on simplicial meshes has recently advanced through the groundbreaking work of Hu, Lin, and Wu (Found. Comput. Math. 24, 2024). Their framework characterizes smoothness via moments of normal derivatives over subsimplices, leading to explicit degrees of freedom and unisolvence, unifying earlier constructions. However, the absence of explicit basis functions has left these spaces largely inaccessible for practical computation. In parallel, multivariate spline theory (Chui and Lai, J. Approx. Theory 60, 1990) enforces $C^m$ smoothness through linear constraints on Bernstein--Bézier coefficients, but stable, locally supported bases remain elusive beyond low dimensions. Building on the geometric decomposition of the simplicial lattice proposed by Chen and Huang (Math. Comp. 93, 2024), this work develops an explicit, computable framework for smooth finite elements. The degrees of freedom defined by moments of normal derivatives are modified to align with the dual basis of the Bernstein polynomials, yielding structured local bases on each simplex. Explicit basis construction is essential not merely for completeness, but for enabling efficient matrix assembly, global continuity, and scalable solution of high-order elliptic partial differential equations. This development closes the gap between theoretical existence and practical realization, making smooth finite element methods accessible to broad computational applications.

math.NA↗

arXiv2025

Discrete Hessian complexes in three dimensions

A family of conforming virtual element Hessian complexes on tetrahedral meshes are constructed based on decompositions of polynomial tensor spaces. They are applied to discretize the linearized time-independent Einstein-Bianchi system with optimal order convergence.

gr-qc↗