任意维微分形式有限元空间中Hodge分解的稳定性

非凸多边形中时变吉恩斯-兰道方程的数学与数值分析（基于霍奇分解）

Mathematical and numerical analysis of time-dependent Ginzburg--Landau equations in nonconvex polygons based on Hodge decomposition

本文研究非凸多边形中时变吉恩斯-兰道方程的well-posed性，通过霍奇分解将解分解为正则与奇异部分，提出分离线性有限元方法并给出误差估计，通过数值实验验证方法的有效性。

一种稳定的收敛方法用于流固相互作用的Hodge分解

A stable and convergent method for Hodge decomposition of fluid-solid interaction

本文提出了一种稳定的收敛方法，用于流固相互作用的Hodge分解，通过扩展的Hodge投影与泊松方程非局部Robin边界条件等价，证明了投影的唯一性和正交性，并展示了数值解的收敛性。

Hodge Decomposition and General Laplacian Solvers for Embedded Simplicial Complexes

We describe a nearly-linear time algorithm to solve the linear system $L_1x = b$ parameterized by the first Betti number of the complex, where $L_1$ is the 1-Laplacian of a simplicial complex $K$ that is a subcomplex of a collapsible complex $X$ linearly embedded in $\mathbb{R}^{3}$. Our algorithm generalizes the work of Black et al.~[SODA2022] that solved the same problem but required that $K$ have trivial first homology. Our algorithm works for complexes $K$ with arbitrary first homology with running time that is nearly-linear with respect to the size of the complex and polynomial with respe…

自适应混合有限元方法对霍奇拉普拉斯方程的收敛性分析：不考虑调和形式

Convergence of Adaptive Mixed Finite Element Methods for the Hodge Laplacian Equation: without harmonic forms

本文基于有限元外 calculus 分析自适应混合有限元方法对霍奇拉普拉斯方程的收敛性，通过霍奇分解、正则分解和有界通配算子获得残差型后验误差估计，并引入额外标记策略确保准正交性，从而在不假设初始网格足够小的情况下实现统一收敛。

A Consistent Discrete 3D Hodge-type Decomposition: implementation and practical evaluation

The Hodge decomposition provides a very powerful mathematical method for the analysis of 2D and 3D vector fields. It states roughly that any vector field can be $L^2$-orthogonally decomposed into a curl-free, divergence-free, and a harmonic field. The harmonic field itself can be further decomposed into three components, two of which are closely tied to the topology of the underlying domain. For practical computations it is desirable to find a discretization which preserves as many aspects inherent to the smooth theory as possible while at the same time remains computationally tractable, in pa…

释放压力：通过离散Helmholtz-Hodge分解的高阶曲面流离散化

Releasing the pressure: High-order surface flow discretizations via discrete Helmholtz-Hodge decompositions

提出一种针对任意拓扑三角曲面上H(div)协调Brezzi-Douglas-Marini有限元的离散Helmholtz-Hodge分解，将无散BDM子空间分解为连续流函数空间的旋转梯度与有限维调和场空间，消除压力和鞍点结构，确保精确切向性、逐点无散性和压力鲁棒性。

Physics-informed Spectral Learning: the Discrete Helmholtz--Hodge Decomposition

In this work, we further develop the Physics-informed Spectral Learning (PiSL) by Espath et al. \cite{Esp21} based on a discrete $L^2$ projection to solve the discrete Hodge--Helmholtz decomposition from sparse data. Within this physics-informed statistical learning framework, we adaptively build a sparse set of Fourier basis functions with corresponding coefficients by solving a sequence of minimization problems where the set of basis functions is augmented greedily at each optimization problem. Moreover, our PiSL computational framework enjoys spectral (exponential) convergence. We regulariz…

环面上的二维离散 Hodge-Dirac 算子

2D discrete Hodge-Dirac operator on the torus

本文基于离散外微分框架，提出环面上二维 Hodge-Dirac 与 Laplace 算子的离散类比，证明离散 Hodge 分解定理，并计算组合环面的上同调群。