Quadratic Discontinuous Galerkin Finite Element Methods for the Unilateral Contact Problem

In this article, we employ discontinuous Galerkin methods for the finite element approximation of the frictionless unilateral contact problem using quadratic finite elements over simplicial triangulation. We first develop a posteriori error estimates in the energy norm wherein, the reliability and efficiency of the proposed a posteriori error estimator is addressed. The suitable construction of the discrete Lagrange multiplier <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>𝝀</m:mi> <m:mi>𝒉</m:mi> </m:msub> </m:math> {{λₕ}} and some intermediate operators play a key role in developing a posteriori error analysis. Further, we establish an optimal a priori error estimates under the appropriate regularity assumption on the exact solution <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>𝒖</m:mi> </m:math> {u} . Numerical results presented on uniform and adaptive meshes illustrate and confirm the theoretical findings.