This paper presents a numerical approach to the stochastic obstacle problem using the stochastic Galerkin (SG) method. Due to the low regularity of the solution, linear finite elements are employed in both the physical and random variable spaces. Properties of random fields and variational inequalities of the first kind are employed to establish the well-posedness of the problem. Finite element spaces are introduced to construct suitable approximation subspaces, and a comprehensive SG formulation is proposed to solve the stochastic obstacle problem. Well-posedness of the discrete formulation is shown and an optimal error estimate for the numerical solution in the H1-norm is derived. Numerical experiments validate the effectiveness of the SG method, showing that the expectation error converges at a rate of O(h) in the H1-norm, consistent with theoretical predictions.

The paper provides a well-posedness analysis for a family of stationary Navier-Stokes-type variational-hemivariational inequalities, motivated by applications in fluid mechanics. The family contains various mixed variational equations, mixed variational inequalities and mixed hemivariational inequalities found in the literature as special cases. The main features of the paper are that the existence and uniqueness of both the velocity field and pressure field are established, and the results are proved in an accessible fashion without the need of knowledge of abstract surjectivity results for pseudomonotone operators as required in many references on variationalhemivariational inequalities. The results are applied to the study of a variational-hemivariational inequality of the Navier-Stokes equations for incompressible fluid flows subject to slip conditions of frictional type, both monotone and non-monotone.

This paper is devoted to the well-posedness analysis of a nonstationary Stokes hemivariational inequality for an incompressible fluid flow described by the Stokes equations subject to a nonsmooth boundary condition of friction type described by the Clarke subdifferential. In a recent paper [19], well-posedness of the nonstationary Stokes hemivariational inequality is studied for both the velocity and pressure fields. The solution existence is shown through a limiting procedure based on temporally semi-discrete approximations for both the velocity and pressure fields. In this paper, a refined well-posedness analysis is provided on the nonstationary Stokes hemivariational inequality under more natural assumptions on the problem data. The solution existence is first shown for the velocity field through a limiting procedure based on temporally semi-discrete approximations of a reduced problem and then the pressure field is recovered with the help of an inf-sup property. In this way, assumptions on the source term and the initial velocity needed in [19] are weakened, and a compatibility condition on initial values of the data is dropped. Moreover, several hemivariational inequalities are introduced for the mathematical model and their equivalence is explored.

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.

In this paper, we first consider the numerical solution of an abstract quasistatic variational inequality arising in the study of quasistatic physical processes. The temporal discretization is carried out by the backward Euler difference scheme, while the spatial discretization is based on the virtual element method. A general framework is provided for the spatially semidiscrete and the fully discrete approximations of the quasistatic variational inequality. Then, as an application of the theoretical results on the abstract problem, a quasistatic contact problem is studied and optimal order error estimates are derived for the lowest-order VEM, under appropriate solution regularity assumptions. Numerical examples are presented to show the performance of the proposed methods.

A nonstationary mixed hemivariational inequality is studied for an incompressible fluid flow described by the Stokes equations subject to a nonsmooth boundary condition of friction type described by the Clarke subdifferential. The solution existence is shown through a limiting procedure based on temporally semi-discrete approximations. Uniqueness of the solution and its continuous dependence on data are also established. Fully discrete numerical methods are introduced to solve the nonstationary mixed hemivariational inequality. The backward Euler scheme is applied to discretize the time derivative, and mixed finite element methods are used for the spatial discretization. An error bound is derived for the numerical solution of the unknown velocity. Numerical results are reported on computer simulations of some examples.

This paper presents a temporal semi-discrete scheme for a nonstationary Stokes hemivariational inequality. To approximate the time derivative, a mid-point discretization is used in the first time step, and the BDF2 for the other time steps. Under appropriate solution regularity assumptions, the scheme is shown to be of second order with respect to the time step-size.

This paper is devoted to studies of doubly-history dependent hemivariational inequalities in contact mechanics. Existence and uniqueness of a solution to the problem is proved by applying a basic well-posedness result combined with a Banach fixed-point argument. A fully discrete scheme is used to solve the problem, with temporal integrals approximated by rectangular rules and the spatial discretization done by the linear element method. Under suitable solution regularity assumptions, an optimal order error bound is proved for the numerical solutions. Finally, simulation results on a numerical example are reported to illustrate numerical convergence orders.

In this paper, we develop a Discontinuous Galerkin (DG) method for solving H(curl)-elliptic hemivariational inequalities. By selecting an appropriate numerical flux, we construct an Interior Penalty Discontinuous Galerkin (IPDG) scheme. A comprehensive numerical analysis of the IPDG method is conducted, addressing key aspects such as consistency, boundedness, stability, and the existence, uniqueness, uniform boundedness of the numerical solutions. Building on these properties, we establish a priori error estimates, demonstrating the optimal convergence order of the numerical solutions under suitable solution regularity assumptions. Finally, a numerical example is presented to illustrate the theoretically predicted convergence order and to show the effectiveness of the proposed method.

This paper provides a well-posedness analysis and a mixed finite element method for a hemivariational inequality of the stationary Navier-Stokes equations with a nonlinear damping term. The Navier-Stokes hemivariational inequality describes a steady incompressible fluid flow subject to a nonsmooth slip boundary condition of friction type. The well-posedness of the Navier-Stokes hemivariational inequality is established by constructing two auxiliary problems and applying Banach fixed point arguments twice. Mixed finite element methods are introduced to solve the problem, and error estimates for the solutions are derived. The error estimates are of optimal order for low-order mixed element pairs under suitable solution regularity assumptions. An efficient iterative algorithm is presented, and numerical results are provided to verify the theoretical analysis.

Morozov's discrepancy principle is commonly adopted in Tikhonov regularization for choosing the regularization parameter. Nevertheless, for a general nonlinear inverse problem, the discrepancy IIF(x(alpha)(delta))-y(delta) IIY does not depend continuously on alpha and it is questionable whether there exists a regularization parameter alpha such that T-1(delta) < ||F(x(delta))-y(delta)||Y < T-2(delta) (1 < T-1 < T-2). In this paper, we prove the existence of alpha under Morozov's discrepancy principle if T2 > (3 + 2-gamma)T-1 or T-1 < T-2/(3 + 2-gamma), where-gamma> 0 is a parameter in a tangential cone condition for the nonlinear operator F. Furthermore, we present results on the convergence of the regularized solutions under Morozov's discrepancy principle. Numerical results are reported on the efficiency of the proposed approach.

A class of evolutionary variational–hemivariational inequalities with a convex constraint is studied in this paper. An inequality in this class involves a first-order derivative and a history-dependent operator. Existence and uniqueness of a solution to the inequality is established by the Rothe method, in which the first-order temporal derivative is approximated by backward Euler’s formula, and the history-dependent operator is approximated by a modified left endpoint rule. The proof of the result relies on basic results in functional analysis only, and it does not require the notion of pseudomonotone operators and abstract surjectivity results for such operators, used in other papers on the Rothe method for other evolutionary variational–hemivariational inequalities. Moreover, a Lipschitz continuous dependence conclusion of the solution on the right-hand side is proved. Finally, a new frictional contact problem for viscoelastic material is discussed, which illustrates an application of the theoretical results.

We explore the well-posedness and conduct a numerical analysis of hemivariational inequalities for the coupled stationary Navier-Stokes/Navier-Stokes system. The interface condition involves the Clark subgradient and serves as a generalization of various interface interaction relations, including nonlinear transmission conditions and friction-type conditions. We present an existence and uniqueness result for a solution of the continuous model. We propose a domain decomposition approach to solve the coupled system and examine the convergence of iterations. Moreover, we use the finite element approximation to discretize the hemivariational inequality of the coupled system and derive error estimates, which lead to an optimal order for the P1 pair under appropriate solution regularity assumptions. Numerical results are reported illustrate the optimal convergence order predicted by theoretical analysis.

We propose three modified contact boundary conditions incorporating both the velocity and the displacement with a parameter δ for the viscoelastic problem. As δ approaches 0, these conditions formally reduce to the conventional Signorini, Tresca-friction, and Clarke-subdifferential type boundary conditions, respectively. Consequently, the modified conditions, as a generalization of the conventional ones, can be viewed as contact conditions in the displacement with a dynamic setting. We derive weak formulations for the viscoelastic contact model under three modified contact conditions and explore their well-posedness. Additionally, we provide bounds on the weak solutions with respect to the parameter δ .

Variational-hemivariational inequalities are an important mathematical framework for nonsmooth problems. The framework can be used to study application problems from physical sciences and engineering that involve non-smooth and even set-valued relations, monotone or non-monotone, among physical quantities. Since no analytic solution formulas are expected for variational-hemivariational inequalities from applications, numerical methods are needed to solve the problems. This paper focuses on numerical analysis of variational-hemivariational inequalities, reporting new results as well as surveying some recent published results in the area. A general convergence result is presented for Galerkin solutions of the inequalities under minimal solution regularity conditions available from the well-posedness theory, and Céa's inequalities are derived for error estimation of numerical solutions. The finite element method and the virtual element method are taken as examples of numerical methods, optimal order error estimates for the linear element solutions are derived when the methods are applied to solve three representative contact problems under certain solution regularity assumptions. Numerical results are presented to show the performance of both the finite element method and the virtual element method, including numerical convergence orders of the numerical solutions that match the theoretical predictions.

Variational-hemivariational inequalities are an area full of interesting and challenging mathematical problems. The area can be viewed as a natural extension of that of variational inequalities. Variational-hemivariational inequalities are valuable for application problems from physical sciences and engineering that involve non-smooth and even set-valued relations, monotone or non-monotone, among physical quantities. In the recent years, there has been substantial growth of research interest in modeling, well-posedness analysis, development of numerical methods and numerical algorithms of variational-hemivariational inequalities. This survey paper is devoted to a brief account of well-posedness and numerical analysis results for variational-hemivariational inequalities. The theoretical results are presented for a family of abstract stationary variational-hemivariational inequalities and the main idea is explained for an accessible proof of existence and uniqueness. To better appreciate the distinguished feature of variational-hemivariational inequalities, for comparison, three mechanical problems are introduced leading to a variational equation, a variational inequality, and a variational-hemivariational inequality, respectively. The paper also comments on mixed variational-hemivariational inequalities, with examples from applications in fluid mechanics, and on results concerning the numerical solution of other types (nonstationary, history dependent) of variational-hemivariational inequalities.

Morozov's discrepancy principle is commonly adopted in Tikhonov regularization for choosing the regularization parameter. Nevertheless, for a general non-linear inverse problem, the discrepancy F(x_α^δ)-y^δ_Y does not depend continuously on α and it is questionable whether there exists a regularization parameter α such that τ_1δ≤F(x_α^δ)-y^δ_Y≤τ_2 δ (1≤τ_1<τ_2). In this paper, we prove the existence of α under Morozov's discrepancy principle if τ_2≥ (3+2γ)τ_1, where γ>0 is a parameter in a tangential cone condition for the nonlinear operator F. Furthermore, we present results on the convergence of the regularized solutions under Morozov's discrepancy principle. Numerical results are reported on the efficiency of the proposed approach.