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

TL;DR

The paper develops a unified framework for the well-posedness and numerical solution of general elliptic mixed variational-hemivariational inequalities, capturing many existing mixed formulations as special cases. It establishes existence, uniqueness, and Lipschitz dependence on data for a rank-$(2,1)$ formulation via a projection-iteration scheme under a smallness condition on nonconvexity parameters, and extends to variational-hemivariational variants and rank-(1,1) degeneracies. A corresponding finite-element discretization is analyzed, yielding a priori error estimates with a residual term and convergence under the discrete inf-sup condition, and a Stokes problem with non-smooth friction on slip boundaries is treated to obtain optimal-order error bounds under regularity. Numerical implementations (Uzawa iterations) demonstrate convergence rates and friction effects, validating the theoretical predictions and showing practical viability for incompressible flows with complex boundary conditions.

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.

Figures (16)

• Figure 1: Triangulation for Example \ref{['sec:ex']}.1
• Figure 2: Velocity field for Example \ref{['sec:ex']}.1
• Figure 3: Pressure isobars for Example \ref{['sec:ex']}.1
• Figure 4: Magnitude of the velocity $|\hbox{\boldmath{$u$}}^{h}|$ for Example \ref{['sec:ex']}.1
• Figure 5: $\hbox{\boldmath{$u$}}_{\tau}$ along $\Gamma_{S,1}$, Example \ref{['sec:ex']}.1

Theorems & Definitions (9)

• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Theorem 2.6
• Theorem 2.8
• Theorem 2.10
• Theorem 3.2
• Corollary 3.3
• Theorem 4.4