On convergence of the penalty method for unilateral contact problems
Provides theoretical justification for the penalty method in contact mechanics, a known numerical technique, but the analysis is incremental.
The paper proves convergence of the penalty method for unilateral contact problems, showing that optimal error estimates are achieved when the penalty parameter is proportional to the mesh size.
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 $\varepsilon = h$.