A class of implicit variational inequalities and applications to frictional contact

This paper deals with the mathematical and numerical analysis of a class of abstract implicit evolution variational inequalities. The results obtained here can be applied to a large variety of quasistatic contact problems in linear elasticity, including unilateral contact or normal compliance conditions with friction. In particular, a quasistatic unilateral contact problem with nonlocal friction is considered. An algorithm is derived and some numerical examples are presented. Copyright © 2009 John Wiley & Sons, Ltd.     

Discontinuous Galerkin finite element methods for variational inequalities of first and second kinds

We develop the error analysis for the h‐version of the discontinuous Galerkin finite element discretization for variational inequalities of first and second kinds. We establish an a priori error estimate for the method which is of optimal order in a mesh dependant as well as L²‐norm.© 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2007     

On continuous first-order methods and their regularized versions for mixed variational inequalities

For mixed variational inequalities in a Hilbert space, we consider continuous first-order methods and obtain sufficient conditions for their strong convergence. If the operator of the problem is not strongly monotone and the functional does not have the property of strong convexity, then regularized versions of these methods are used for the solution of a mixed variational inequality. For the case in which the data are given approximately, we prove the strong convergence of the regularized methods to a normal solution of the original problem. The construction of all methods uses the resolvent of the maximal monotone operator. We obtain sufficient conditions for the unique solvability of the Cauchy problems determining the considered methods.     

Family of perturbation methods for variational inequalities

In recent years, the so-called auxiliary problem principle has been used to derive many iterative type algorithms for solving optimal control, mathematical programming, and variational inequality problems. In the present paper, we use this principle in conjunction with the epiconvergence theory to introduce and study a general family of perturbation methods for solving nonlinear variational inequalities over a product space of reflexive Banach spaces. We do not assume that the monotone operator involved in our general variational inequality problem is of potential type. Several known iterative algorithms, which can be obtained from our theory, are also discussed.This work was completed while the second author was visiting the Department of Mathematics of the University of Washington, Seattle, Washington under financial support from the Belgian Fonds National de la Recherche Scientifique, Grant FNRS: B8/5-JS-9. 549.     

Application of penalty methods to non-stationary variational inequalities

We solve a general variational inequality problem in a finite-dimensional setting, where only approximation sequences are known instead of exact values of the cost mapping and feasible set. We suggest to utilize a sequence of solutions of auxiliary problems based on a penalty method. Its convergence is attained without concordance of penalty and approximation parameters under mild coercivity type conditions. We also show that the regularized version of the penalty method enables us to further weaken the coercivity condition.     

Subdifferential inclusions and quasi-static hemivariational inequalities for frictional viscoelastic contact problems

We survey recent results on the mathematical modeling of nonconvex and nonsmooth contact problems arising in mechanics and engineering. The approach to such problems is based on the notions of an operator subdifferential inclusion and a hemivariational inequality, and focuses on three aspects. First we report on results on the existence and uniqueness of solutions to subdifferential inclusions. Then we discuss two classes of quasi-static hemivariational ineqaulities, and finally, we present ideas leading to inequality problems with multivalued and nonmonotone boundary conditions encountered in mechanics.     

Convergence of direct methods for paramonotone variational inequalities

We analyze one-step direct methods for variational inequality problems, establishing convergence under paramonotonicity of the operator. Previous results on the method required much more demanding assumptions, like strong or uniform monotonicity, implying uniqueness of solution, which is not the case for our approach.     

Mixed projection methods for systems of variational inequalities

Let H be a real Hilbert space. Let be bounded and continuous mappings where D(F) and D(K) are closed convex subsets of H. We introduce and consider the following system of variational inequalities: find [u ^*,v ^*]∈D(F) × D(K) such that This system of variational inequalities is closely related to a pseudomonotone variational inequality. The well-known projection method is extended to develop a mixed projection method for solving this system of variational inequalities. No invertibility assumption is imposed on F and K. The operators K and F also need not be defined on compact subsets of H.      

Finite volume method for the variational inequalities of first and second kinds

We propose and analyze the finite volume method for solving the variational inequalities of first and second kinds. The stability and convergence analysis are given for this method. For the elliptic obstacle problem, we derive the optimal error estimate in the H¹‐norm. For the simplified friction problem, we establish an abstract H¹‐error estimate, which implies the convergence if the exact solution u∈H¹(Ω) and the optimal error estimate if u∈H^{1 + α}(Ω),0 < α≤2. Copyright © 2015 John Wiley & Sons, Ltd.     

Projection methods for equations of the second kind

A class of projection methods, differing from the classical projection methods, is studied for the equation y = f + Ky, where K is a compact linear operator in a Banach space E, and f?E, In these methods K is approximated by a finite-rank operator K_n, which is constructed with the aid of certain projection operators, and which satisfies K_nz = Kz for all z belonging to a chosen subspace U_n ? E. Under certain conditions, it is shown that the convergence of the approximate solution is faster than that of any classical projection method based on the subspace U_n. In an example, U_n is taken to consist of piecewise constant functions, and the projections are so chosen that the method becomes equivalent to a single iteration of a classical method, the collocation method; in this case the error (in the supremum norm) is $\text{O(}\text{1}\text{n}{}^2\text{)}$, compared with $\text{O(}\text{1}\text{n}\text{)}$ for the collocation method.     

Applications of fixed-point methods to discrete variational and quasi-variational inequalities

Summary In this paper, discrete analogues of variational inequalities (V.I.) and quasi-variational inequalities (Q.V.I.), encountered in stochastic control and mathematical physics, are discussed.It is shown that those discrete V.I.'s and Q.V.I.'s can be written in the fixed point formx=Tx such that eitherT or some power ofT is a contraction. This leads to globally convergent iterative methods for the solution of discrete V.I.'s and Q.V.I.'s, which are very suitable for implementation on parallel computers with single-instruction, multiple-data architecture, particularly on massively parallel processors (M.P.P.'s).This research is in part supported by the U.S. Department of Energy, Engineering Research Program, under Contract No. DE-AS05-84EH13145     

Regularization and iterative methods for monotone inverse variational inequalities

We consider the monotone inverse variational inequality: find $x\in H$ such that $$\begin{aligned} f(x)\in \Omega , \quad \left\langle \tilde{f}-f(x),x\right\rangle \ge 0, \quad \forall \tilde{f}\in \Omega , \end{aligned}$$ where $\Omega $ is a nonempty closed convex subset of a real Hilbert space $H$ and $f:H\rightarrow H$ is a monotone mapping. A general regularization method for monotone inverse variational inequalities is shown, where the regularizer is a Lipschitz continuous and strongly monotone mapping. Moreover, we also introduce an iterative method as discretization of the regularization method. We prove that both regularized solution and an iterative method converge strongly to a solution of the inverse variational inequality.     

Uzawa iteration method for stokes type variational inequality of the second kind

In this paper,the Uzawa iteration algorithm is applied to the Stokes problem with nonlinear slip boundary conditions whose variational formulation is the variational inequality of the second kind.Firstly, the multiplier in a convex set is introduced such that the variational inequality is equivalent to the variational identity.Moreover,the solution of the variational identity satisfies the saddle-point problem of the Lagrangian functional ?.Subsequently,the Uzawa algorithm is proposed to solve the solution of the saddle-point problem. We show the convergence of the algorithm and obtain the convergence rate.Finally,we give the numerical results to verify the feasibility of the Uzawa algorithm.     

Penalty methods for a system of constrained variational inequalities

In this paper, using the penalty method in conjunction with graph-convergence, we study the existence of solutions for a class of generalized variational inequalities with variational problem constraints. Our results extend, improve and develop some known results in this field. Our method of proofs is very simple and does not use the hemicontinuity nor the pseudo-continuity properties of monotone operators.     

Descent methods for a class of generalized variational inequalities

In this paper we propose a class of differentiable gap functions in order to formulate a generalized variational inequality (GVI) problem, involving a set-valued map with closed and convex graph, as an optimization problem. We also show that under appropriate assumptions on the set-valued map, any stationary point of the equivalent optimization problem is a global optimal solution and solves the GVI. Finally, we describe descent methods for solving the optimization problem equivalent to the GVI and we prove its global convergence.     

A class of projection-contraction methods applied to monotone variational inequalities

First, an extension of the projection-contraction (PC) method is introduced, which generalizes a class of the existing PC methods, and then the extended projection-contraction (EPC) method is applied to the solvability of a class of general monotone variational inequalities.     

Entropic proximal decomposition methods for convex programs and variational inequalities

We consider convex optimization and variational inequality problems with a given separable structure. We propose a new decomposition method for these problems which combines the recent logarithmic-quadratic proximal theory introduced by the authors with a decomposition method given by Chen-Teboulle for convex problems with particular structure. The resulting method allows to produce for the first time provably convergent decomposition schemes based on C ^∞ Lagrangians for solving convex structured problems. Under the only assumption that the primal-dual problems have nonempty solution sets, global convergence of the primal-dual sequences produced by the algorithm is established. Received: October 6, 1999 / Accepted: February 2001?Published online September 17, 2001