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.

翻译：本文采用间断伽辽金（DG）方法，基于单纯形网格上的二次有限元，对无摩擦单边接触问题进行有限元逼近。首先，在精确解$\b{u}$的适当正则性假设下，建立了最优先验误差估计。进一步，在DG范数下分析后验误差估计，其中讨论了所提后验误差估计子的可靠性和效率。离散拉格朗日乘子$\b{\lambda_h}$的合理构造及若干中间算子在后验误差分析中起关键作用。在均匀网格和自适应网格上呈现的数值结果验证并确认了理论发现。