Quadratic Discontinuous Galerkin methods for Unilateral Contact Problem

TL;DR

This work develops quadratic discontinuous Galerkin (DG) methods for the frictionless unilateral Signorini contact problem, combining a priori and a posteriori analyses on simplicial meshes. By formulating the problem as a variational inequality and discretizing with a DG space, the authors prove optimal a priori convergence in the DG norm under $u\in H^{2+\epsilon}(\Omega)$, and construct a residual-based a posteriori error estimator in the DG norm that is both reliable and (locally) efficient. A discrete Lagrange multiplier $\boldsymbol{\lambda}_h$ is defined on a suitable edge-based space, with sign properties that facilitate the error analysis, and an enriching map $E_h$ links DG and conforming spaces to handle nonconformity. Numerical experiments on uniform and adaptive meshes for SIPG and NIPG verify the theoretical convergence rates and demonstrate effective adaptivity driven by the estimator. The results advance robust DG-based discretizations for variational inequalities arising in contact mechanics and provide practical tools for error-controlled simulations in unilateral contact problems.

Abstract

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 $\b{λ_h}$ 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.

Figures (7)

• Figure 2.1: Physical setting of Signorini problem
• Figure 7.1: Convergence of error and estimator for SIPG and NIPG method for Model Problem 1.
• Figure 7.2: Error estimator contributions for SIPG and NIPG method for Model Problem 1.
• Figure 7.3: Plot of efficiency index for SIPG and NIPG method for Model Problem 1.
• Figure 7.4: Convergence of estimator for SIPG and NIPG method for Model Problem 2.

Theorems & Definitions (26)

• Lemma 2.1
• Remark 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 4.1
• Lemma 4.2
• proof : Proof of Theorem \ref{['thm1']}
• Lemma 5.1
• proof