On convergence of the penalty method for unilateral contact problems

TL;DR

This paper analyzes the convergence of the penalty method for unilateral contact problems, establishing conditions under which the method converges as the penalty parameter and discretization size tend to zero.

Contribution

It provides a comprehensive convergence analysis for both continuous and finite element discretized penalty methods in unilateral contact problems.

Findings

Convergence of the continuous penalty solution as the penalty parameter approaches zero.

Finite element penalty method error estimates match those of the constrained problem when  = h.

Validation of the penalty method's effectiveness in unilateral contact problems.

Abstract

We present a convergence analysis of the penalty method applied to unilateral contact problems in two and three space dimensions. We first consider, under various regularity assumptions on the exact solution to the unilateral contact problem, the convergence of the continuous penalty solution as the penalty parameter $\varepsilon$ vanishes. Then, the analysis of the finite element discretized penalty method is carried out. Denoting by $h$ the discretization parameter, we show that the error terms we consider give the same estimates as in the case of the constrained problem when the penalty parameter is such that $\varepsilon = h$.