Long Chen

Long Chen, Jun Hu, Xuehai Huang

In this paper, we design two classes of stabilized mixed finite element methods for linear elasticity on simplicial grids. In the first class of elements, we use $\boldsymbol{H}(\mathbf{div}, Ω; \mathbb{S})$-$P_k$ and $\boldsymbol{L}^2(Ω; \mathbb{R}^n)$-$P_{k-1}$ to approximate the stress and displacement spaces, respectively, for $1\leq k\leq n$, and employ a stabilization technique in terms of the jump of the discrete displacement over the faces of the triangulation under consideration; in the second class of elements, we use $\boldsymbol{H}_0^1(Ω; \mathbb{R}^n)$-$P_{k}$ to approximate the displacement space for $1\leq k\leq n$, and adopt the stabilization technique suggested by Brezzi, Fortin, and Marini. We establish the discrete inf-sup conditions, and consequently present the a priori error analysis for them. The main ingredient for the analysis is two special interpolation operators, which can be constructed using a crucial $\boldsymbol{H}(\mathbf{div})$ bubble function space of polynomials on each element. The feature of these methods is the low number of global degrees of freedom in the lowest order case. We present some numerical results to demonstrate the theoretical estimates.

Chen Long, Wenxiao Zhang, Ruihui Li et al.

Point cloud upsampling is to densify a sparse point set acquired from 3D sensors, providing a denser representation for the underlying surface. Existing methods divide the input points into small patches and upsample each patch separately, however, ignoring the global spatial consistency between patches. In this paper, we present a novel method PC$^2$-PU, which explores patch-to-patch and point-to-point correlations for more effective and robust point cloud upsampling. Specifically, our network has two appealing designs: (i) We take adjacent patches as supplementary inputs to compensate the loss structure information within a single patch and introduce a Patch Correlation Module to capture the difference and similarity between patches. (ii) After augmenting each patch's geometry, we further introduce a Point Correlation Module to reveal the relationship of points inside each patch to maintain the local spatial consistency. Extensive experiments on both synthetic and real scanned datasets demonstrate that our method surpasses previous upsampling methods, particularly with the noisy inputs. The code and data are at \url{https://github.com/chenlongwhu/PC2-PU.git}.

Long Chen, Jianguo Huang

Some error analysis on virtual element methods including inverse inequalities, norm equivalence, and interpolation error estimates are presented for polygonal meshes which admits a virtual quasi-uniform triangulation.