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In this paper, we provide an alternative approach to establish the solution existence and uniqueness for elliptic variational-hemivariational inequalities. The new approach is based on elementary results from functional analysis, and thus removes the need of the notion of pseudomonotonicity and the dependence on surjectivity results for pseudomonotone operators. This makes the theory of elliptic variational-hemivariational inequalities more accessible to applied mathematicians and engineers. In addition, equivalent minimization principles are further explored for particular elliptic variational-hemivariational inequalities. Representative examples from contact mechanics are discussed to illustrate application of the theoretical results.

Several discontinuous Galerkin (DG) methods are introduced for solving a frictional contact problem with normal compliance, which is modeled as a quasi-variational inequality. Consistency, boundedness, and stability are established for the DG methods. Two numerical examples are presented to illustrate the performance of the DG methods.

In this paper, we consider numerical approximations of a contact problem in rate-type viscoplasticity. The contact conditions are described in term of a subdifferential and include as special cases some classical frictionless boundary conditions. The contact problem consists of an evolution equation coupled with a time-dependent variational inequality. Error estimates for both spatially semi-discrete and fully discrete solutions are derived and some convergence results are shown. Under appropriate regularity assumptions on the exact solution, error estimates are obtained.

The Kacanov method is an iteration method for solving nonlinear Problems. In this paper, we apply the Kacanov method to solve the problem of quasi-Newtonian flow obeying the Carreau law. The emphasis of the paper is the derivation of a posteriori error estimate for Kacanov iterates. A numerical example shows the efficiency of the error estimate.

In this paper, we consider a frictional contact problem with normal compliance. The problem is formulated as a variational inequality, and is shown to possess a unique solution. The finite element approximation is analyzed, with a Ceelike inequality proved. A major difficulty in actually solving the problem is caused by the presence of a non—differentiable functional in the formulation of the problem. Here we discuss the regularization method to overcome the difficulty. A posteriori error estimates are derived

We develop an a-posteriori analysis for errors caused by material idealizations in modeling one-dimensional elastostatic problems. Two kinds of material idealizations are considered. A main feature of the error analysis given here is that we only need the solution of the idealized problem to estimate the modeling error. We also discuss how to compute quantitative error bounds when the idealized problem is solved by the finite element method. An example is given showing the effectiveness of the a-posteriori error estimates.

The best possible constant in the trace inequality is shown to be given by a quantity in terms of the solution of an elliptic boundary value problem, where, Ω ⊂ RN is a Lipschitz domain, Γ1 is a measurable subset of ∂Ω, and k > 0 is fixed.