arXiv subjects

Kaibo Hu

Publications and source records attributed to Kaibo Hu.

At least 19 recordsLinked to original sources

arXiv2026

Analytical properties of polygonal Stokes and BGG Hessian complexes with application to Kirchhoff--Love plates

We develop and analyse a new arbitrary-order method for the Kirchhoff--Love plate problem on general polygonal meshes. The construction relies on a discrete Hessian complex obtained through the Bernstein--Gelfand--Gelfand construction, which stacks a Stokes complex on top of a tensorised de Rham complex. A key ingredient of the analysis is a detailed study of the former. We specifically establish primal and adjoint consistency estimates, as well as uniform Poincar\'e inequalities, and use these results to derive the analytical properties of the resulting discrete Hessian complex. The proposed method is then shown to be coercive and to converge with order $k+1$ with respect to the mesh size, where $k\ge 0$ is the polynomial degree of the complex. Numerical experiments on polygonal meshes confirm the theoretical convergence rates.

math.NA↗

arXiv2026

ReLU$^k$ Neural de Rham Complexes

We construct finite-dimensional de Rham subcomplexes generated by fixed-neuron shallow ReLU$^k$ neural networks, a class of spaces known to provide optimal approximation rates. For neurons of the form $s_i(x)=\omega_i\cdot x+b_i$, we introduce spaces of neural differential forms: differential $p$-forms whose coefficients are the ReLU$^k$ ridge functions $\sigma_{k-p}(s_i)$. These spaces are compatible with the exterior derivative because differentiating a ReLU power lowers its order by one, and for each fixed neuron, differentiation amounts to exterior multiplication by the fixed one-form $ d s_i$. Under a linear independence assumption on the lowest-order family $\{\sigma_{k-d}(s_i)\}_{i=1}^n$, the global complex decomposes into independent neuron-wise Koszul complexes. We prove exactness in arbitrary dimension and provide a geometric sufficient condition for the required linear independence. Numerical experiments based on the resulting complex provide evidence of stable discretizations and of convergence rates consistent with the underlying approximation theory, and exhibit no spurious modes in eigenvalue problems considered.

math.NA↗

arXiv2026

Optimal block preconditioners for a mass-conserving mixed stress formulation of Stokes flow

We present optimal block diagonal and triangular preconditioners for a mass-conserving mixed stress formulation of Stokes flow. The algebraic formulation leads to a double saddle point system with unknowns corresponding to discrete stress, velocity, vorticity, and pressure. MINRES equipped with a block diagonal preconditioner for an augmented Lagrangian formulation of this system is analyzed and shown to be optimal, in the sense that the convergence rate is independent of key parameters such as mesh size and kinematic viscosity. GMRES equipped with a block triangular preconditioner is also analyzed using a field-of-values approach. Finally, we present numerical results for both two- and three-dimensional model problems to validate the parameter robustness of the proposed preconditioners.

math.NA↗

arXiv2026

Global and local helicity-preservation in the finite element discretization of magnetic relaxation

Magnetic relaxation drives plasma toward lower-energy equilibria under helicity constraints. In ideal magnetohydrodynamics (MHD), helicity is locally conserved, while resistive theories such as Taylor relaxation preserve only global helicity. This distinction has important implications for structure-preserving numerical methods. We compare three finite element formulations: an unconstrained scheme that does not conserve helicity, a mixed method based on finite element exterior calculus that preserves discrete local helicity on magnetically closed subdomains, and a Lagrange multiplier approach that enforces only global helicity conservation. Numerical experiments with magnetic knots and braids show that helicity-based constraints provide effective topological barriers when the relevant helicity-type invariant is nonzero, but do not fully characterize braided field-line topology when it vanishes. These results clarify both the strengths and the possible limitations of helicity-based structure-preserving finite element methods for magnetic relaxation.

math.NA↗

arXiv2026

Finite element methods for isometric embedding of Riemannian manifolds

The isometric embedding problem for Riemannian manifolds, which connects intrinsic and extrinsic geometry, is a central question in differential geometry with deep theoretical significance and wide-ranging applications. Despite extensive analytical progress, the nonlinear and degenerate nature of this problem has hindered the development of rigorous numerical analysis in this area. As the first step toward addressing this gap, we study the numerical approximation of Weyl's problem, i.e., the isometric embedding of two-dimensional Riemannian manifolds with positive Gaussian curvature into $\mathbb{R}^3$, by establishing a new weak formulation that naturally leads to a numerical scheme well suited for high-order finite element discretization, and conducting a systematic analysis to prove the well-posedness of this weak formulation, the existence and uniqueness of its numerical solution, as well as its convergence with error estimates. This provides a foundational framework for computing isometric embeddings of Riemannian manifolds into Euclidean space, with the goal of extending it to a broader range of cases and applications in the future. Our framework also extends naturally to the isometric embedding of the Ricci flow, with rigorous error estimates, enabling the visualization of geometric evolutions in intrinsic curvature flows. Numerical experiments support the theoretical analysis by demonstrating the convergence of the method and its effectiveness in simulating isometric embeddings of given Riemannian manifolds as well as Ricci flows.

math.NA↗

arXiv2025

Hodge-Dirac wave systems and structure-preserving discretizations of the linearized Einstein equations

We derive a reformulation of the linearized Arnowitt-Deser-Misner (ADM) equations as a Hodge-Dirac wave system with the divdiv complex, addressing challenges in numerical relativity such as gauge fixing, constraint propagation, and tensor symmetries. The differential and algebraic structures of the divdiv complex ensure the well-posedness of the formulation and facilitate structure-preserving discretization via finite element exterior calculus. We establish the well-posedness of this Hodge-Dirac wave equation and develop a discretization scheme applicable to both conforming and non-conforming discrete complexes, deriving error estimates under minimal assumptions.

gr-qc↗

arXiv2025

Design and homological analysis of twisted and BGG Stokes-de Rham complexes on polygonal meshes

We design a discrete Bernstein--Gelfand--Gelfand (BGG) diagram in a two-dimensional setting on polygonal meshes based on the DDR framework; the diagram is made of a discrete Stokes polygonal complex and a tensorised Discrete de Rham complex, and the BGG construction leads to novel twisted and Hessian complexes applicable on generic polygonal meshes; such complexes enable, in particular, discretisations of plate problems. Complete homological properties of the discrete Stokes complex and of the complexes built from the BGG diagram are established.

math.NA↗

arXiv2025

A 2-complex containing Sobolev spaces of matrix fields

Using a generalization of complexes, called 2-complexes, this paper defines and analyzes new Sobolev spaces of matrix fields and their interrelationships within a commuting diagram. These spaces have very weak second-order derivatives. An example is the space of matrix fields of square-integrable components whose row-wise divergence followed by yet another divergence operation yield a function in a standard negative-order Sobolev space. Similar spaces where the double divergence is replaced by a curl composed with divergence, or a double curl operator (the incompatibility operator), are also studied. Stable decompositions of such spaces in terms of more regular component functions (which are continuous in natural norms) are established. Appropriately ordering such Sobolev spaces with and without boundary conditions (in a weak sense), we discover duality relationships between them. Motivation to study such Sobolev spaces, from a finite element perspective and implications for weak well-posed variational formulations are pointed out.

math.AP↗

arXiv2025

Convergence and Stability of Discrete Exterior Calculus for the Hodge Laplace Problem in Two Dimensions

We prove convergence and stability of the discrete exterior calculus (DEC) solutions for the Hodge-Laplace problems in two dimensions for families of meshes that are non-degenerate Delaunay and shape regular. We do this by relating the DEC solutions to the lowest order finite element exterior calculus (FEEC) solutions. A Poincar\'e inequality and a discrete inf-sup condition for DEC are part of this proof. We also prove that under appropriate geometric conditions on the mesh the DEC and FEEC norms are equivalent. Only one side of the norm equivalence is needed for proving stability and convergence and this allows us to relax the conditions on the meshes.

math.NA↗

arXiv2025

Many facets of cohomology: Differential complexes and structure-aware formulations

Complexes and cohomology, traditionally central to topology, have emerged as fundamental tools across applied mathematics and the sciences. This survey explores their roles in diverse areas, from partial differential equations and continuum mechanics to reformulations of the Einstein equations and network theory. Motivated by advances in compatible and structure-preserving discretisation such as Finite Element Exterior Calculus (FEEC), we examine how differential complexes encode critical properties such as existence, uniqueness, stability and rigidity of solutions to differential equations. We demonstrate that various fundamental concepts and models in solid and fluid mechanics are essentially formulated in terms of differential complexes.

math.NA↗

arXiv2025

Finite element form-valued forms: Construction

We provide a finite element discretization of $\ell$-form-valued $k$-forms on triangulation in $\mathbb{R}^{n}$ for general $k$, $\ell$ and $n$ and any polynomial degree. The construction generalizes finite element Whitney forms for the de~Rham complex and their higher-order and distributional versions, the Regge finite elements and the Christiansen--Regge elasticity complex, the TDNNS element for symmetric stress tensors, the MCS element for traceless matrix fields, the Hellan--Herrmann--Johnson (HHJ) elements for biharmonic equations, and discrete divdiv and Hessian complexes in [Hu, Lin, and Zhang, 2025]. The construction discretizes the Bernstein--Gelfand--Gelfand (BGG) diagrams. Applications of the construction include discretization of strain and stress tensors in continuum mechanics and metric and curvature tensors in differential geometry in any dimension.

math.NA↗

arXiv2025

Helicity-preserving finite element discretization for magnetic relaxation

The Parker conjecture, which explores whether magnetic fields in perfectly conducting plasmas can develop tangential discontinuities during magnetic relaxation, remains an open question in astrophysics. Helicity conservation provides a topological barrier during relaxation, preventing topologically nontrivial initial data relaxing to trivial solutions; preserving this mechanism discretely over long time periods is therefore crucial for numerical simulation. This work presents an energy- and helicity-preserving finite element discretization for the magneto-frictional system for investigating the Parker conjecture. The algorithm preserves a discrete version of the topological barrier and a discrete Arnold inequality. We also propose extensions of the notion of helicity and the Arnold inequality to certain kinds of topologically nontrivial domains. Numerical experiments demonstrate that helicity preservation is crucial in obtaining physically meaningful simulations of magnetic relaxation, providing an example where structure-preserving schemes are necessary.

math.NA↗

arXiv2024

Intrinsic mixed finite element methods for linear Cosserat elasticity

We propose two parameter-robust mixed finite element methods for linear Cosserat elasticity. The Cosserat coupling constant $\mu_c$, connecting the displacement $u$ and rotation vector $\omega$, leads to possible locking phenomena in finite element methods. The formal limit of $\mu_c\to\infty$ enforces the constraint $\frac{1}{2}\operatorname{curl} u = \omega$ and leads to the fourth-order couple stress problem. Viewing the linear Cosserat model as the Hodge-Laplacian problem of a twisted de~Rham complex, we derive structure-preserving distributional finite element spaces, where the limit constraint is fulfilled in the discrete setting. Applying the mass conserving mixed stress (MCS) method for the rotations, the resulting scheme is robust in $\mu_c$. Combining it with the tangential-displacement normal-normal-stress (TDNNS) method for the displacement part, we obtain additional robustness in the nearly incompressible regime and for anisotropic structures. Using a post-processing scheme for the rotations, we prove optimal convergence rates independent of the Cosserat coupling constant $\mu_c$. We demonstrate the performance of the proposed methods in several numerical benchmark examples.

math.NA↗

arXiv2024

Uniformly $hp$-stable elements for the elasticity complex

For the discretization of symmetric, divergence-conforming stress tensors in continuum mechanics, we prove inf-sup stability bounds which are uniform in polynomial degree and mesh size for the Hu--Zhang finite element in two dimensions. This is achieved via an explicit construction of a bounded right inverse of the divergence operator, with the crucial component being the construction of bounded Poincar\'e operators for the stress elasticity complex which are polynomial-preserving, in the Bernstein--Gelfand--Gelfand framework of the finite element exterior calculus. We also construct $hp$-bounded projection operators satisfying a commuting diagram property and $hp$-stable Hodge decompositions. Numerical examples are provided.

math.NA↗

arXiv2023

Quadratic and cubic Lagrange finite elements for mixed Laplace eigenvalue problems on criss-cross meshes

In [6], it was shown that the linear Lagrange element space on criss-cross meshes and its divergence exhibit spurious eigenvalues when applied in the mixed formulation of the Laplace eigenvalue problem, despite satisfying both the inf-sup condition and ellipticity on the discrete kernel. The lack of a Fortin interpolation is responsible for the spurious eigenvalues produced by the linear Lagrange space. In contrast, results in [8] confirm that quartic and higher-order Lagrange elements do not yield spurious eigenvalues on general meshes without nearly singular vertices, including criss-cross meshes as a special case. In this paper, we investigate quadratic and cubic Lagrange elements on criss-cross meshes. We prove the convergence of discrete eigenvalues by fitting the Lagrange elements on criss-cross meshes into a complex and constructing a Fortin interpolation. As a by-product, we construct bounded commuting projections for the finite element Stokes complex, which induces isomorphisms between cohomologies of the continuous and discrete complexes. We provide numerical examples to validate the theoretical results.

math.NA↗

arXiv2023

Extended Regge complex for linearized Riemann-Cartan geometry and cohomology

We show that the cohomology of the Regge complex in three dimensions is isomorphic to $\mathcal{H}^{\scriptscriptstyle \bullet}_{dR}(Ω)\otimes\mathcal{RM}$, the infinitesimal-rigid-body-motion-valued de~Rham cohomology. Based on an observation that the twisted de~Rham complex extends the elasticity (Riemannian deformation) complex to the linearized version of coframes, connection 1-forms, curvature and Cartan's torsion, we construct a discrete version of linearized Riemann-Cartan geometry on any triangulation and determine its cohomology.

math.NA↗

arXiv2023

Distributional Hessian and divdiv complexes on triangulation and cohomology

In this paper, we construct discrete versions of some Bernstein-Gelfand-Gelfand (BGG) complexes, i.e., the Hessian and the divdiv complexes, on triangulations in 2D and 3D. The sequences consist of finite elements with local polynomial shape functions and various types of Dirac measure on subsimplices. The construction generalizes Whitney forms (canonical conforming finite elements) for the de Rham complex and Regge calculus/finite elements for the elasticity (Riemannian deformation) complex from discrete topological and Discrete Exterior Calculus perspectives. We show that the cohomology of the resulting complexes is isomorphic to the continuous versions, and thus isomorphic to the de~Rham cohomology with coefficients.

math.NA↗

arXiv2023

Finite elements for symmetric and traceless tensors in three dimensions

We construct a family of finite element sub-complexes of the conformal complex on tetrahedral meshes and show their exactness on contractible domains. This complex includes vector fields and symmetric and traceless tensor fields, connected through the conformal Killing operator, the linearized Cotton-York operator, and the divergence operator, respectively. This leads to discrete versions of transverse traceless (TT) tensors, i.e., symmetric, traceless and divergence-free matrix fields, in continuum mechanics and general relativity. We also show the inf-sup stability of the $H(\operatorname{div})$-conforming finite element symmetric and traceless tensors paired with discontinuous vectors.

math.NA↗