A new penalty method for elliptic quasivariational inequalities

TL;DR

This paper introduces a new penalty method for elliptic quasivariational inequalities, providing convergence criteria, theoretical analysis, and numerical validation in elastic contact problems.

Contribution

The paper develops a novel penalty approach for elliptic quasivariational inequalities, establishing convergence conditions and applying them to elastic contact problems with numerical demonstrations.

Findings

Convergence of penalty solutions to the original problem under certain conditions

Application of the method to elastic frictional contact problems

Numerical simulations confirming theoretical convergence results

Abstract

We consider a class of elliptic quasivariational inequalities in a reflexive Banach space $X$ for which we recall a convergence criterion obtained in [10]. Each inequality $\cal P$ in the class is governed by a set of constraints $K$ and has a unique solution $u\in K$. The criterion provides necessary and sufficient conditions which guarantee that an arbitrary sequence $\{u_n\}\subset X$ converges to the solution $u$. Then, we consider a sequence $\{\cal P_n\}$ of unconstrained variational-hemivariational inequalities governed by a sequence of parameters $\{\lambda_n\}\subset\mathbb{R}_+$. We use our criterion to deduce that, if for each $n\in\mathbb{N}$ the term $u_n$ represents a solution of Problem $\cal P_n$, then the sequence $\{u_n\}$ converges to $u$ as $\lambda_n\to 0$. We apply our abstract results in the study of an elastic frictional contact problem with unilateral constraints and provide the corresponding mechanical interpretations. We also present numerical simulation in the study of a two-dimensional example which represents an evidence of our convergence results.