Using adjoint error estimation techniques for elastohydrodynamic lubrication line contact problems

The use of an adjoint technique for goal‐based error estimation described by Hartit et al. (Int. J. Numer. Meth. Fluids 2005; 47 :1069–1074) is extended to the numerical solution of free boundary problems that arise in elastohydrodynamic lubrication (EHL). EHL systems are highly nonlinear and consist of a thin‐film approximation of the flow of a non‐Newtonian lubricant which separates two bodies that are forced together by an applied load, coupled with a linear elastic model for the deformation of the bodies. A finite difference discretization of the line contact flow problem is presented, along with the numerical evaluation of an exact solution for the elastic deformation, and a moving grid representation of the free boundary that models cavitation at the outflow in this one‐dimensional case. The application of a goal‐based error estimate for this problem is then described. This estimate relies on the solution of an adjoint problem; its effectiveness is demonstrated for the physically important goal of the total friction through the contact. Finally, the application of this error estimate to drive local mesh refinement is demonstrated. Copyright © 2010 John Wiley & Sons, Ltd.     

Boltzmann-Poisson系统解的动量及能量估计

In this paper, the properties of the solutions of Boltzmann-Poisson system in three dimensions are considered. The estimates of the energies (the kinetic and the potential energies) and the γ-th moments with γ∈[2,3) are obtained. The latter answers an open problem.     

Anisotropic Crouzeix-Raviart Type Nonconforming Finite Element Methods to Variational Inequality Problem with Displacement Obstacle

In this paper, anisotropic Crouzeix-Raviart type nonconforming finite element methods are considered for solving the second order variational inequality with displacement obstacle. The convergence analysis is presented and the optimal order error estimates are obtained under the hypothesis of the finite length of the free boundary. Numerical results are provided to illustrate the correctness of theoretical analysis.     

Numerical Approximations of the Spectral Discretization of Flame Front Model

In this paper, we consider the numerical solution of the flame front equation, which is one of the most fundamental equations for modeling combustion theory. A schema combining a finite difference approach in the time direction and a spectral method for the space discretization is proposed. We give a detailed analysis for the proposed schema by providing some stability and error estimates in a particular case. For the general case, although we are unable to provide a rigorous proof for the stability, some numerical experiments are carried out to verify the efficiency of the schema. Our numerical results show that the stable solution manifolds have a simple structure when $\beta$ is small, while they become more complex as the bifurcation parameter $\beta$ increases. At last numerical experiments were performed to support the claim the solution of flame front equation preserves the same structure as K-S equation.     

沪深股市相关结构之谜:基于贝叶斯Copula的研究

目前沪深股市相关结构的Copula模型选择差异很大,并没有形成统一的认识。在指出现有Copula检验要受到模型参数估计影响后,引入了贝叶斯估计方法将模型参数估计与拟合优度检验有效的分开。接着,沪深股市相关性的贝叶斯实证结果发现两市相关结构Copula模型具有时变特征,势必导致当前研究结果的不一致;同时也反映了Copula对样本区间选择有很强的依赖性。     

Recovery Type a Posteriori Error Estimates of Fully Discrete Finite Element Methods for General Convex Parabolic Optimal Control Problems

This paper is concerned with recovery type a posteriori error estimates of fully discrete finite element approximation for general convex parabolic optimal control problems with pointwise control constraints. The time discretization is based on the backward Euler method. The state and the adjoint state are approximated by piecewise linear functions and the control is approximated by piecewise constant functions. We derive the superconvergence properties of finite element solutions. By using the superconvergence results, we obtain recovery type a posteriori error estimates. Some numerical examples are presented to verify the theoretical results.     

Spectral Method for Nonlinear Stochastic Partial Differential Equations of Elliptic Type

This paper is concerned with the numerical approximations of semi-linear stochastic partial differential equations of elliptic type in multi-dimensions. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral method. Numerical results demonstrate the good performance of the spectral method.     

Long time existence of Ricci-harmonic flow

We give a survey about recent results on Ricci-harmonic flow.     

On $L^2$-Stability Analysis of Time-Domain Acoustic Scattering Problems with Exact Nonreflecting Boundary Conditions

This paper is devoted to stability analysis of the acoustic wave equation exterior to a bounded scatterer, where the unbounded computational domain is truncated by the exact time-domain circular/spherical nonreflecting boundary condition (NRBC). Different from the usual energy method, we adopt an argument that leads to $L^2$-a priori estimates with minimum regularity requirement for the initial data and source term. This needs some delicate analysis of the involved NRBC. These results play an essential role in the error analysis of the interior solvers (e.g., finite-element/spectral- element/spectral methods) for the reduced scattering problems. We also apply the technique to analyze a time-domain waveguide problem.     

二阶非线性椭圆型方程于无界域上的斜微商问题

在机械和物理中有许多问题的数学模型是一、二阶非线性椭圆型方程于包含无穷远点的多连通域上的某些边值问题，该文讨论了二阶非线性椭圆型方程于包含无穷远点的多连通域上的斜微商边值问题．