Numerical analysis of stationary variational-hemivariational inequalities with applications in contact mechanics

This paper is devoted to numerical analysis of general finite element approximations to stationary variational-hemivariational inequalities with or without constraints. The focus is on convergence under minimal solution regularity and error estimation under suitable solution regularity assumptions that cover both internal and external approximations of the stationary variational-hemivariational inequalities. A framework is developed for general variational-hemivariational inequalities, including a convergence result and a Céa type inequality. It is illustrated how to derive optimal order error estimates for linear finite element solutions of sample problems from contact mechanics.

[1]  Panagiotis D. Panagiotopoulos,et al.  Hemivariational Inequalities: Applications in Mechanics and Engineering , 1993 .

[2]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[3]  R. S. Falk.  Error estimates for the approximation of a class of variational inequalities , 1974 .

[4]  Philippe G. Ciarlet,et al.  The finite element method for elliptic problems , 2002, Classics in applied mathematics.

[5]  Mircea Sofonea,et al.  Numerical Analysis of Elliptic Hemivariational Inequalities , 2017, SIAM J. Numer. Anal..

[6]  Dumitru Motreanu,et al.  Nonsmooth Variational Problems and Their Inequalities , 2007 .

[7]  M. Sofonea,et al.  Mathematical Models in Contact Mechanics , 2012 .

[8]  Jaroslav Haslinger,et al.  Finite Element Method for Hemivariational Inequalities , 1999 .

[9]  Wenbin Chen,et al.  Numerical Analysis of History-dependent Variational-hemivariational Inequalities , 2020, ArXiv.

[10]  Zdzisław Denkowski,et al.  An Introduction to Nonlinear Analysis: Theory , 2013 .

[11]  Weimin Han,et al.  Numerical Analysis of a Hyperbolic Hemivariational Inequality Arising in Dynamic Contact , 2015, SIAM J. Numer. Anal..

[12]  Krzysztof Bartosz,et al.  Numerical Methods for Evolution Hemivariational Inequalities , 2015 .

[13]  Mircea Sofonea,et al.  Nonlinear Inclusions and Hemivariational Inequalities: Models and Analysis of Contact Problems , 2012 .

[14]  W. Han,et al.  Quasistatic Contact Problems in Viscoelasticity and Viscoplasticity , 2002 .

[15]  R. Glowinski,et al.  Numerical Analysis of Variational Inequalities , 1981 .

[16]  W. Han,et al.  Advances in Variational and Hemivariational Inequalities: Theory, Numerical Analysis, and Applications , 2015 .

[17]  I. Ekeland,et al.  Convex analysis and variational problems , 1976 .

[18]  Diana Baader,et al.  Theoretical Numerical Analysis A Functional Analysis Framework , 2016 .

[19]  Mircea Sofonea,et al.  A Class of Variational-Hemivariational Inequalities in Reflexive Banach Spaces , 2017 .

[20]  Dumitru Motreanu,et al.  Minimax Theorems and Qualitative Properties of the Solutions of Hemivariational Inequalities , 1998 .

[21]  F. Clarke.  Optimization And Nonsmooth Analysis , 1983 .

[22]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[23]  Mircea Sofonea,et al.  A Class of Variational-Hemivariational Inequalities with Applications to Frictional Contact Problems , 2014, SIAM J. Math. Anal..

[24]  J. Oden,et al.  Contact Problems in Elasticity: A Study of Variational Inequalities and Finite Element Methods , 1987 .

[25]  P. D. Panagiotopoulos,et al.  Mathematical Theory of Hemivariational Inequalities and Applications , 1994 .

[26]  Mircea Sofonea,et al.  Advances in Variational and Hemivariational Inequalities , 2015 .