Numerical analysis for a new kind of obstacle problem

In this paper, we consider a new kind of obstacle problem for the elastic membrane. The obstacle is made of a rigid body covered by a soft layer that is deformable and allows penetration. It assigns a reactive normal pressure, which depends on the interpenetration of the membrane and the obstacle, during the contact process. Three equivalent descriptions of the new obstacle problem are derived, namely the energy form, the variational inequality form and the differential equation form. The existence and uniqueness of the solution are proved. Based on the variational inequality form, we derive an optimal order error estimate for the finite element approximate solution under appropriate solution regularity assumptions. We also introduce a series of penalized problems and prove its convergence result as the penalty parameter converges to infinity. Numerical examples are reported on using linear elements to solve the new obstacle problem, and the simulation results are in good agreement with the theoretical analysis