Numerical analysis of history-dependent variational-hemivariational inequalities
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https://doi.org/10.1007/s11425-019-1672-4
. ARTICLES .
Numerical analysis of history-dependent variational-hemivariational inequalities Shufen Wang1 , Wei Xu2 , Weimin Han3 & Wenbin Chen1,∗ 1School
of Mathematical Sciences, Fudan University, Shanghai 200433, China; of Science, Tongji Zhejiang College, Jiaxing 314051, China; 3Department of Mathematics, University of Iowa, Iowa City, IA 52242, USA 2Faculty
Email: [email protected], [email protected], [email protected], [email protected] Received August 29, 2019; accepted April 3, 2020
Abstract
In this paper, numerical analysis is carried out for a class of history-dependent variational-
hemivariational inequalities by arising in contact problems. Three different numerical treatments for temporal discretization are proposed to approximate the continuous model. Fixed-point iteration algorithms are employed to implement the implicit scheme and the convergence is proved with a convergence rate independent of the time step-size and mesh grid-size. A special temporal discretization is introduced for the history-dependent operator, leading to numerical schemes for which the unique solvability and error bounds for the temporally discrete systems can be proved without any restriction on the time step-size. As for spatial approximation, the finite element method is applied and an optimal order error estimate for the linear element solutions is provided under appropriate regularity assumptions. Numerical examples are presented to illustrate the theoretical results. Keywords
variational-hemivariational inequality, Clarke subdifferential, history-dependent operator, fixed-
point iteration, optimal order error estimate, contact mechanics MSC(2010)
47J20, 65N30, 65N15, 74M15
Citation: Wang S F, Xu W, Han W M, et al. Numerical analysis of history-dependent variational-hemivariational inequalities. Sci China Math, 2020, 63, https://doi.org/10.1007/s11425-019-1672-4
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Introduction
The theory of variational and hemivariational inequalities plays an important role in the study of nonlinear problems arising in Contact Mechanics, Physics, Economics and Engineering. It is generally agreed that interest in variational inequalities started with a contact problem posed by Signorini in 1930s. The mathematical theory of variational inequalities relies on the properties of monotonicity, convexity and the subdifferential of a convex function. The existence and uniqueness results can be found in [3, 17, 18]. In terms of the numerical analysis for variational inequalities, the readers are referred to, e.g., [7, 8, 15]. Hemivariational inequalities as a useful generalization of variational inequalities were introduced in early 1980s by Panagiotopoulos [22]. For hemivariational inequalities, the notion of the subdifferential in the sense of Clarke [5,6], defined for locally Lipschitz function, plays an important role. Mathematical theory of hemivariational inequalities is documented in several research monographs (see, e.g., [4, 19, 21, 23, 27]). * Corresponding author c Scie
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