A First-order Two-scale Analysis for Contact Problems with Small Periodic Configurations

TL;DR

This paper develops a homogenization and computational framework for linear elliptic contact problems with small periodic coefficients, providing error estimates and numerical validation.

Contribution

It introduces a first-order two-scale analysis for contact problems with periodic coefficients, including homogenization results and error estimates for finite element methods.

Findings

Homogenization of contact problems with periodic coefficients achieved.

Error estimates of $O(\epsilon^{1/2})$ and $O(\epsilon)$ derived.

Numerical experiments confirm theoretical error bounds.

Abstract

This paper is devoted to studying a type of contact problems modeled by hemivariational inequalities with small periodic coefficients appearing in PDEs, and the PDEs we considered are linear, second order and uniformly elliptic. Under the assumptions, it is proved that the original problem can be homogenized, and the solution weakly converges. We derive an $O(\epsilon^{1/2})$ estimation which is pivotal in building the computational framework. We also show that Robin problems--- a special case of contact problems, it leads to an $O(\epsilon)$ estimation in $L^2$ norm. Our computational framework is based on finite element methods, and the numerical analysis is given, together with experiments to convince the estimation.