No abstract

In this paper, we consider the numerical method for an optimal control problem governed by an obstacle problem. An approximate optimization problem is proposed by regularizing the original non-differentiable constrained problem with a simple method. The connection between the two formulations is established through some convergence results. A sufficient condition is derived to decide whether a solution of the first-order optimality system is a global minimum. The method with a second-order in time dissipative system is developed to solve the optimality system numerically. Several numerical examples are reported to show the effectiveness of the proposed method.

In this paper, we consider a new obstacle problem for the elastic membrane lying above an elastic-rigid obstacle. The elastic-rigid obstacle allows limited penetration and offers a nonmonotone reactive force. We introduce the mathematical model and prove that its weak form, which is a variational-hemivariational inequality, has a unique solution. Then, we consider a discrete scheme to solve the problem. The optimal-order error estimate under appropriate regularity assumptions is derived. Finally, numerical examples are reported, from which the theoretical predicted optimal-order error estimate can be clearly observed.

This paper is devoted to a study of a new kind of bilateral obstacle problem. The obstacles are made of rigid bodies covered by soft layers which are deformable and allow penetration. A model of an elastic rigid bilateral obstacle problem is established, and three equivalent descriptions are derived: the energy form, the variational inequality form, and the differential equation form. We prove the solution existence and uniqueness of this model and provide an error estimate of the numerical solutions. The optimal order error estimate for the linear finite element method is derived under the proper regularity assumption. A penalty method is introduced to solve the finite element approximation problem, and the convergence results are obtained when the penalty parameter tends to infinity. Several numerical examples are reported, and the results are in good agreement with the theoretical analysis.