Analysis of a new kind of elastic-rigid bilateral obstacle problem

Abstract

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.