Chouly and Hild consider the contact problem of an elastic 2D and 3D body. Small strains are assumed and the contact is a straight line segment in 2D and a polygon in 3D. They reformulate the variational inequality using the penalty method, and prove the existence of a unique solution to the weak formulation of the problem for any small positive parameter resulting from the penalty method. The paper proves a bound in several norms on the consistency error introduced by the penalty method. The rate of convergence depends on the norm used.

The authors also determine the order of convergence when the mesh size and the penalty parameter both tend to zero. In the case where the two are the same, this error estimate matches that of the finite element approximation of the variational inequality. This last result shows the equivalency of the two approximate methods in terms of rate of convergence.

Convergence analysis is one of the fundamental pillars of computational science and engineering during a design process. Thus, the work presented here ensures that the numerical solution of nonlinear contact problems has a solid theoretical foundation.