Optimal error estimates of the unilateral contact problem in a curved and smooth boundary domain by the penalty method

We consider linear finite elements to approximate the elasticity equations with unilateral contact boundary conditions, in a bounded two- or three-dimensional domain with curved and smooth boundary. We use the penalty method to weakly impose these boundary conditions. We establish an error estimate in the energy norm with respect to the mesh size $h$ and the penalty parameter $\varepsilon $. Assuming $\boldsymbol{H}^{\frac{3}{2}+\nu }\left (\varOmega \right )$ regularity of the solution, $0 < \nu \leq \frac{1}{2}$, we obtain an $\mathcal{O}\,(h^{\frac{1}{2}+\nu } + \varepsilon ^{\frac{1}{2}+\nu })$ convergence rate. Therefore, if the penalty parameter is chosen as $\varepsilon (h) := ch^{\theta }$ with $0 < \theta \leq 1$, we obtain an $\mathcal{O}\,(h^{\theta (\frac{1}{2}+\nu )})$ convergence rate. Thus, the optimal linear convergence rate is obtained when $\varepsilon $ behaves like $h$ (that is, $\theta = 1$) and $\nu = \frac{1}{2}$. We present a numerical example to illustrate the theoretical analysis.