Well-posedness and Numerical Analysis of Mixed Variational-hemivariational Inequalities

TL;DR

This paper investigates the well-posedness and numerical solutions of a broad class of mixed variational-hemivariational inequalities, providing error analysis and applying results to fluid flow problems with slip conditions.

Contribution

It introduces a unified framework for analyzing mixed variational-hemivariational inequalities and develops stable finite element methods with proven convergence rates.

Findings

Established well-posedness using projection iteration.

Derived optimal error estimates for finite element approximations.

Numerical results confirm theoretical convergence orders.

Abstract

The paper is devoted to well-posedness analysis and the numerical solution of a family of general elliptic mixed variational-hemivariational inequalities. Various mixed variational equations, mixed variational inequalities and mixed hemivariational inequalities found in the literature are special cases of the mixed variational-hemivariational inequalities. Well-posedness of the mixed variational-hemivariational inequalities and their numerical approximations are studied via the projection iteration technique. Error analysis of the numerical methods is presented. The results are applied to the study of a variational-hemivariational inequality of the Stokes equations for incompressible fluid flows subject to slip conditions of frictional type, both monotone and non-monotone. Optimal order error estimates are derived for the use of some stable finite element space pairs under certain solution regularity assumptions. Numerical results are reported demonstrating the theoretical prediction of convergence orders.