Some Error Analysis on Virtual Element Methods

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

A robust lower order mixed finite element method for a strain gradient elasticity model

A robust nonconforming mixed finite element method is developed for a strain gradient elasticity (SGE) model. In two and three dimensional cases, a lower order $C^0$-continuous $H^2$-nonconforming finite element is constructed for the displacement field through enriching the quadratic Lagrange element with bubble functions. This together with the linear Lagrange element is exploited to discretize a mixed formulation of the SGE model. The robust discrete inf-sup condition is established. The sharp and uniform error estimates with respect to both the small size parameter and the Lam\'{e} coefficient are achieved, which is also verified by numerical results. In addition, the uniform regularity of the SGE model is derived under two reasonable assumptions.Comment: 25 page

TorchCP: A Library for Conformal Prediction based on PyTorch

TorchCP is a Python toolbox for conformal prediction research on deep learning models. It contains various implementations for posthoc and training methods for classification and regression tasks (including multi-dimension output). TorchCP is built on PyTorch (Paszke et al., 2019) and leverages the advantages of matrix computation to provide concise and efficient inference implementations. The code is licensed under the LGPL license and is open-sourced at $\href{https://github.com/ml-stat-Sustech/TorchCP}{\text{this https URL}}$

On the Local Discontinuous Galerkin Method for Linear Elasticity

Following Castillo et al. (2000) and Cockburn (2003), a general framework of constructing discontinuous Galerkin (DG) methods is developed for solving the linear elasticity problem. The numerical traces are determined in view of a discrete stability identity, leading to a class of stable DG methods. A particular method, called the LDG method for linear elasticity, is studied in depth, which can be viewed as an extension of the LDG method discussed by Castillo et al. (2000) and Cockburn (2003). The error bounds in L2-norm, H1-norm, and a certain broken energy norm are obtained. Some numerical results are provided to confirm the convergence theory established

Vibration analysis of Kirchhoff plates by the Morley element method

AbstractVibration analysis of Kirchhoff plates is of great importance in many engineering fields. The semi-discrete and the fully discrete Morley element methods are proposed to solve such a problem, which are effective even when the region of interest is irregular. The rigorous error estimates in the energy norm for both methods are established. Some reasonable approaches to choosing the initial functions are given to keep the good convergence rate of the fully discrete method. A number of numerical results are provided to illustrate the computational performance of the method in this paper