基于对偶理论的椭圆变分不等式的后验误差分析  

A Posteriori Error Analysis for Elliptic Variational Inequalities Basedon Duality Theory

作　　者：何莉敏[1,2] 王娟 侯玉双[1,3] HE Li-min;WANG Juan;HOU Yu-shuang(School of Science,Inner Mongolia University of Science and Technology,Baotou 014010;School of Mathematics and Statistics,Xi'an Jiaotong University,Xi'an 710049;School of Physical Science and Technology,Southwest University,Chongqing 400715)

机构地区：[1]内蒙古科技大学理学院,包头014010 [2]西安交通大学数学与统计学院,西安710049 [3]西南大学物理科学与技术学院,重庆400715

出　　处：《工程数学学报》2020年第3期370-390,共21页Chinese Journal of Engineering Mathematics

基　　金：国家自然科学基金(11801287,61663035);内蒙古自治区自然科学基金(2018BS01002,2018MS06017,NJZZ18140).

摘　　要：本文基于对偶理论对椭圆变分不等式的正则化方法提供一个相对全面的后验误差分析.我们分别考虑了摩擦接触问题和障碍问题,通过选取一种不同形式的有界算子和泛函,推导了其对偶形式并给出了正则化方法的H^1范后验误差估计.最后,利用凸分析中的对偶理论建立了障碍问题的残量型后验误差估计的一般框架.同时我们选取一种特殊的对偶变量和泛函形式得到该问题的残量型误差估计及其有效性.数值解的后验误差估计是发展有效自适应算法的基础而模型误差的后验误差估计在分析问题中数据的不确定影响时是非常有用的.In this paper,we provide a relatively complete a posteriori error analysis for the regularization method via duality theory for elliptic variational inequalities.The model problems considered in the paper are a friction contact problem and an obstacle problem,respectively.Choosing a different bounded operator form and a functional form,we perform their dual formations and give an H1-norm a posteriori error estimation based on the regularization method which is usually used in solving non-differentiable minimization problems.A posteriori error estimates,with residual type for an obstacle problem in the general framework,is established by using duality theory in convex analysis.At the same time,we make a particular choice of the dual variable that leads to a residual-based error estimate of the model problem and its efficiency.A posteriori error estimates for numerical solutions are the basis for developing efficient adaptive algorithms,whereas a posteriori estimates for modeling errors are useful for analyzing the effects of uncertainties in problem data on the solution.

关 键 词：后验误差估计 正则化方法 椭圆变分不等式 有效性 对偶理论 

分 类 号：O178[理学—数学]