A Posteriori Error Estimation For Signorini Problem

The finite element method is an active research field in computational mathematics.As an efficient numerical approach, it has been widely used for solving a variety ofdifferential equation in the past five decades. Based on a posteriori error estimate, theadaptive finite element method improves the computing efficiency. It achieves highaccuracy with lower memory usage and less time consuming. Adaptive finite elementmethod becomes an effective numerical method for solving many mathematical andphysical problems.Variational inequalities form an important family of nonlinear problem, which havebeen widely used in physical, engineering, finance, and management science. The famousSignorini problem is a representative of elliptic variational inequality of the first kind,which describes a frictionless contact between an elastic body and a rigid object. Thisthesis is intended to apply the adaptive finite element method to solve Signorini problem.We investigate a new approach for analyzing the realibility and efficiency of the aposteriori error estimators. Comparing with other previous papers discussing this problem,the approach in this thesis has two advantages. First, this method is simple; another, it canderive not only the residual error estimators, but also can be extended to other type of aposteriori error estimators.The idea we used in this paper is inspired by the article [17]. We extend the ideatherein to analyze the efficiency and realibility of a posteriori error estimator for Signoriniproblem. The idea is to transform an inequality problem to a differential boundary valueproblem, then apply a posteriori error theory of the linear elliptic differential equation. Ofcourse, the diffculty still exists, and we must analyze carefully about some key points. Inthis thesis, we derive reliable a posteriori error estimators, and the efficiency of theestimators is theoretically explored.In chapter1, we review some theoretical foundation which is necessary for thefollowing analysis, including variational inequalities, finite element method and adaptivealgotirhm. In chapter2, firstly, we introduce physical background of Signorini problem,and then give ghe mathematical representation of the simplified Signorini problem and its discrete form with finite element method. Then in chapter3, we study a posteriori errorestimator for the linear finite element method of Signorini problem, and analyze therealibity and efficiency of the estimator. It is the key part of this thesis. We make asurmmary of this thesis in the last chapter, and then propose the study direction for thefuture.