Yizhou Liang

Pablo D. Brubeck, Yizhou Liang, Charles Parker

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.

Yizhou Liang, Ngoc Tien Tran

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.

Nengbo Lu, Minghua Pan, Shaohua Sun et al.

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.