Quadratic Discontinuous Galerkin methods for Unilateral Contact Problem

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 $\mathit{\text{a}}\mathit{\text{priori}}$ error estimates under the appropriate regularity assumption on the exact solution $\text{b}u$. Further, we analyze $\mathit{\text{a}}\mathit{\text{posteriori}}$ error estimates in the DG norm wherein, the reliability and efficiency of the proposed $\mathit{\text{a}}\mathit{\text{posteriori}}$ error estimator is addressed. The suitable construction of discrete Lagrange multiplier $\text{b}{\lambda }_h$ and some intermediate operators play a key role in developing $\mathit{\text{a}}\mathit{\text{posteriori}}$ error analysis. Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings.