arXiv subjects

Yizhou Liang

Publications and source records attributed to Yizhou Liang.

15 recordsLinked to original sources

arXiv2026

Finite Element Complexes with Trace Structures: A unified framework for cohomology and bounded interpolation

This paper develops a unified framework for studying the cohomology and constructing $L^2$-bounded commuting interpolation operators for finite element complexes, such as the Hessian, divdiv, elasticity and Stokes complexes. Compared with the standard de Rham complex, these complexes present additional challenges arising from algebraic constraints on tensor fields, such as symmetry and tracelessness, as well as additional smoothness requirements. The main ingredient of our framework is a direct sum decomposition of finite element complexes into skeletal and bubble parts. To establish this decomposition, we introduce two key structures: trace structures and generalized currents. In this decomposition, the bubble part is characterized by bubble complexes associated with subsimplices of different dimensions, including vertices, edges, and faces. We prove that the exactness of the resulting bubble complexes ensures that the finite element complex has the correct cohomology. Based on this decomposition, we further construct $L^2$-bounded interpolation operators for these finite element complexes.

math.NA↗

arXiv2026

High-order guaranteed lower bounds for the Schrödinger eigenvalue problem

We compute unconditional guaranteed lower eigenvalue bounds for Schrödinger operators using a modified hybrid high-order method with built-in correction for the potential defect. The method assumes exact numerical integration and allows possibly unbounded potentials in $L^p$ for $p>n/2$ and $p\geq2$ on general polyhedral domains. We prove that the high-order correction preserves optimal a priori convergence orders, and derive reliable and efficient a posteriori error estimators. Numerical experiments provide striking evidence of optimal high-order convergence rates of the adaptive algorithm.

math.NA↗

arXiv2026

Finite element exterior calculus for spectra and pseudospectra of advection-diffusion of differential forms

Numerical investigations of dynamo action have remained active in fluid mechanics over the past decades, presenting numerous challenges and open problems. Meanwhile, the development of structure-preserving methods and finite element exterior calculus (FEEC) inspires a revisit of numerical dynamo studies and an exploration of existing open questions in computation. In this paper, we present a FEEC approach for dynamo problems. In particular, we investigate structure-preserving finite element schemes for computing the spectra and pseudospectra of advection-diffusion operators of differential forms. The schemes and their analysis are based on finite element de Rham complexes.

math.NA↗

arXiv2026

Cohomology of Finite Element Stokes Complexes on Alfeld Splits

We show that the cohomology of the finite element Stokes complex consisting of piecewise polynomials spaces on an Alfeld split mesh from Fu, Guzmán, & Neilan (2020, Math. Comp., 89, 1059--1091) is isomorphic to the cohomologies of the continuous Stokes and de Rham complexes. We also construct novel "minimal" conforming finite element complexes where the $H^1$-conforming space is the lowest-order space from Guzmán & Neilan (2018, SIAM J. Numer. Anal., 56, 2826--2844) and the $L^2$-conforming space is piecewise constants. These minimal complexes also have cohomologies isomorphic to the continuous Stokes and de Rham complexes. We further construct local, bounded, cochain projections for the minimal complexes. All the results hold for strongly Lipschitz domains with nontrivial topologies and in the presence of mixed boundary conditions.

math.NA↗

arXiv2026

A hybrid high-order method for the biharmonic problem

This paper proposes a new hybrid high-order discretization for the biharmonic problem and the corresponding eigenvalue problem. The discrete ansatz space includes degrees of freedom in $n-2$ dimensional submanifolds (e.g., nodal values in 2D and edge values in 3D), in addition to the typical degrees of freedom in the mesh and on the hyperfaces in the HHO literature. This approach enables the characteristic commuting property of the hybrid high-order methodology in any space dimension. The main results are guaranteed lower eigenvalue bounds of higher order. Furthermore, we derive quasi-best approximation estimates as well as reliable and efficient a~posteriori error estimators under minimal regularity assumptions on the exact solution. The latter motivates an adaptive mesh-refining algorithm that empirically recovers optimal convergence rates for singular solutions.

math.NA↗

arXiv2026

GS-DMSR: Dynamic Sensitive Multi-scale Manifold Enhancement for Accelerated High-Quality 3D Gaussian Splatting

In the field of 3D dynamic scene reconstruction, how to balance model convergence rate and rendering quality has long been a critical challenge that urgently needs to be addressed, particularly in high-precision modeling of scenes with complex dynamic motions. To tackle this issue, this study proposes the GS-DMSR method. By quantitatively analyzing the dynamic evolution process of Gaussian attributes, this mechanism achieves adaptive gradient focusing, enabling it to dynamically identify significant differences in the motion states of Gaussian models. It then applies differentiated optimization strategies to Gaussian models with varying degrees of significance, thereby significantly improving the model convergence rate. Additionally, this research integrates a multi-scale manifold enhancement module, which leverages the collaborative optimization of an implicit nonlinear decoder and an explicit deformation field to enhance the modeling efficiency for complex deformation scenes. Experimental results demonstrate that this method achieves a frame rate of up to 96 FPS on synthetic datasets, while effectively reducing both storage overhead and training time.Our code and data are available at https://anonymous.4open.science/r/GS-DMSR-2212.

cs.CV↗

arXiv2025

A Hybrid High-Order Method for the Gross--Pitaevskii Eigenvalue Problem

We introduce a hybrid high-order method for approximating the ground state of the nonlinear Gross--Pitaevskii eigenvalue problem. Optimal convergence rates are proved for the ground state approximation, as well as for the associated eigenvalue and energy approximations. Unlike classical conforming methods, which inherently provide upper bounds on the ground state energy, the proposed approach gives rise to guaranteed and asymptotically exact lower energy bounds. Importantly, and in contrast to previous works, they are obtained directly without the need of post-processing, leading to more accurate guaranteed lower energy bounds in practice.

math.NA↗

arXiv2024

Positivity preserving finite element method for the Gross-Pitaevskii ground state: discrete uniqueness and global convergence

We propose a positivity preserving finite element discretization for the nonlinear Gross-Pitaevskii eigenvalue problem. The method employs mass lumping techniques, which allow to transfer the uniqueness up to sign and positivity properties of the continuous ground state to the discrete setting. We further prove that every non-negative discrete excited state up to sign coincides with the discrete ground state. This allows one to identify the limit of fully discretized gradient flows, which are typically used to compute the discrete ground state, and thereby establish their global convergence. Furthermore, we perform a rigorous a priori error analysis of the proposed non-standard finite element discretization, showing optimal orders of convergence for all unknowns. Numerical experiments illustrate the theoretical results of this paper.

math.NA↗

arXiv2024

Mixed finite elements for the Gross-Pitaevskii eigenvalue problem: a priori error analysis and guaranteed lower energy bound

We establish an a priori error analysis for the lowest-order Raviart-Thomas finite element discretisation of the nonlinear Gross-Pitaevskii eigenvalue problem. Optimal convergence rates are obtained for the primal and dual variables as well as for the eigenvalue and energy approximations. In contrast to conformal approaches, which naturally imply upper energy bounds, the proposed mixed discretisation provides a guaranteed and asymptotically exact lower bound for the ground state energy. The theoretical results are illustrated by a series of numerical experiments.

math.NA↗

arXiv2023

Local Bounded Commuting Projection Operators for Discrete Gradgrad Complexes

This paper discusses the construction of local bounded commuting projections for discrete subcomplexes of the gradgrad complexes in two and three dimensions, which play an important role in the finite element theory of elasticity (2D) and general relativity (3D). The construction first extends the local bounded commuting projections to the discrete de Rham complexes to other discrete complexes. Moreover, the argument also provides a guidance in the design of new discrete gradgrad complexes.

math.NA↗

arXiv2023

Local Bounded Commuting Projection Operator for Discrete de Rham Complexes

The local bounded commuting projection operators of nonstandard finite element de Rham complexes in two and three dimensions are constructed systematically. The assumptions of the main result are mild and can be verified. For three dimensions, the result can be applied to the standard finite element de Rham complex, Hermite complex, Argyris complex and Neilan's Stokes complex. For two dimensions, the result can be applied to the Hermite--Stenberg complex and the Falk--Neilan Stokes complex.

math.NA↗

arXiv2023

Finite element grad grad complexes and elasticity complexes on cuboid meshes

This paper constructs two conforming finite element grad grad and elasticity complexes on the cuboid meshes. For the finite element grad grad complex, an $H^2$ conforming finite element space, an $\boldsymbol{H}(\operatorname{curl}; \mathbb{S})$ conforming finite element space, an $\boldsymbol{H}(\operatorname{div}; \mathbb{T})$ conforming finite element space and an $\boldsymbol{L}^2$ finite element space are constructed. Further, a finite element complex with reduced regularity is also constructed, whose degrees of freedom for the three diagonal components are coupled. For the finite element elasticity complex, a vector $\boldsymbol{H}^1$ conforming space and an $\boldsymbol{H}(\operatorname{curl}\operatorname{curl}^{T}; \mathbb{S})$ conforming space are constructed. Combining with an existing $\boldsymbol{H}(\operatorname{div}; \mathbb{S}) \cap \boldsymbol{H}(\operatorname{div}\operatorname{div}; \mathbb{S})$ element and $\boldsymbol{H}(\operatorname{div}; \mathbb{S})$ element, respectively, these finite element spaces form two different finite element elasticity complexes. The exactness of all the finite element complexes is proved.

math.NA↗

arXiv2022

New conforming finite element divdiv complexes in three dimensions

In this paper, the first family of conforming finite element divdiv complexes on cuboid grids in three dimensions is constructed. Besides, a new family of conforming finite element divdiv complexes with enhanced smoothness on tetrahedral grids is presented. These complexes are exact in the sense that the range of each discrete map is the kernel space of the succeeding one.

math.NA↗

arXiv2021

Conforming finite element DIVDIV complexes and the application for the linearized Einstein-Bianchi system

This paper presents the first family of conforming finite element divdiv complexes on tetrahedral grids in three dimensions. In these complexes, finite element spaces of $H(\text{divdiv},Ω;\mathbb{S})$ are from a current preprint [Chen and Huang, arXiv: 2007.12399, 2020] while finite element spaces of both $H(\text{symcurl},Ω;\mathbb{T})$ and $H^1(Ω;\mathbb{R}^3)$ are newly constructed here. It is proved that these finite element complexes are exact. As a result, they can be used to discretize the linearized Einstein-Bianchi system within the dual formulation.

math.NA↗

arXiv2020

Conforming Discrete Gradgrad-Complexes in Three Dimensions

In this paper, the first family of conforming discrete three dimensional Gradgrad-complexes consisting of finite element spaces is constructed. These discrete complexes are exact in the sense that the range of each discrete map is the kernel space of the succeeding one. These spaces can be used in the mixed form of the linearized Einstein-Bianchi system.

math.NA↗