  In this article, we employ discontinuous Galerkin (DG) methods for the finite
element approximation of the frictionless unilateral contact problem using
quadratic finite elements over simplicial triangulation. We first establish an
optimal \textit{a priori} error estimates under the appropriate regularity
assumption on the exact solution $\b{u}$. Further, we analyze \textit{a
posteriori} error estimates in the DG norm wherein, the reliability and
efficiency of the proposed \textit{a posteriori} error estimator is addressed.
The suitable construction of discrete Lagrange multiplier $\b{\lambda_h}$ and
some intermediate operators play a key role in developing \textit{a posteriori}
error analysis. Numerical results presented on uniform and adaptive meshes
illustrate and confirm the theoretical findings.
