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Pointwise A posteriori error control of quadratic Discontinuous Galerkin Methods for the unilateral contact problem

Rohit Khandelwal, Kamana Porwal, Tanvi Wadhawan

TL;DR

This paper develops a pointwise a posteriori error estimator for the quadratic discontinuous Galerkin discretization of the Signorini unilateral contact problem on polygonal domains. The approach combines barrier functionals for the continuous solution, a discrete contact force density, and Green's matrix estimates for divergence-type operators to derive a reliable and efficient estimator in the supremum norm. The framework includes an enriched, smoothing-enriched analysis, a discrete Lagrange multiplier formulation, and a detailed reliability and efficiency theory, complemented by two 2D numerical experiments that demonstrate refined mesh concentration near the contact boundary and confirm optimal convergence behavior. The results enhance adaptive refinement strategies for unilateral contact problems, enabling accurate pointwise error control in practical computations.

Abstract

An a posteriori error bound for the pointwise error of the quadratic discontinuous Galerkin method for the unilateral contact problem on polygonal domain is presented. The pointwise a posteriori error analysis is based on the direct use of a priori estimates of the Green's matrix for the divergence type operators and the suitable construction of the discrete contact force density $\bσ_h$ and barrier functions for the continuous solution. Several numerical experiments (in two dimension) are presented to illustrate the reliability and efficiency properties of the proposed aposteriori error estimator.

Pointwise A posteriori error control of quadratic Discontinuous Galerkin Methods for the unilateral contact problem

TL;DR

This paper develops a pointwise a posteriori error estimator for the quadratic discontinuous Galerkin discretization of the Signorini unilateral contact problem on polygonal domains. The approach combines barrier functionals for the continuous solution, a discrete contact force density, and Green's matrix estimates for divergence-type operators to derive a reliable and efficient estimator in the supremum norm. The framework includes an enriched, smoothing-enriched analysis, a discrete Lagrange multiplier formulation, and a detailed reliability and efficiency theory, complemented by two 2D numerical experiments that demonstrate refined mesh concentration near the contact boundary and confirm optimal convergence behavior. The results enhance adaptive refinement strategies for unilateral contact problems, enabling accurate pointwise error control in practical computations.

Abstract

An a posteriori error bound for the pointwise error of the quadratic discontinuous Galerkin method for the unilateral contact problem on polygonal domain is presented. The pointwise a posteriori error analysis is based on the direct use of a priori estimates of the Green's matrix for the divergence type operators and the suitable construction of the discrete contact force density and barrier functions for the continuous solution. Several numerical experiments (in two dimension) are presented to illustrate the reliability and efficiency properties of the proposed aposteriori error estimator.
Paper Structure (12 sections, 15 theorems, 106 equations, 8 figures)

This paper contains 12 sections, 15 theorems, 106 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Continuous Problem
  3. Quadratic Discontinuous Galerkin FEM
  4. Discrete Lagrange Multiplier
  5. The Smoothing Operator
  6. A Posteriori Error Estimates
  7. Existence of Green's matrix
  8. Reliability of the error estimator
  9. Barrier Functions for the continuous solution
  10. Bound on $\|\boldsymbol{u}^{corr}\|_{\boldsymbol{L}^{\infty}(\bar{ \Omega}) }$
  11. Efficiency of the error estimator
  12. Numerical Experiments

Key Result

Lemma 2.2

Let $\boldsymbol{u}$ be the solution of the continuous variational inequality eq:CVI. Then, the following relations hold

Figures (8)

  • Figure 6.1: Physical setting of Model Problem 1.
  • Figure 6.2: Plot of convergence of Error and Estimator for SIPG and NIPG methods for Model Problem 1.
  • Figure 6.3: Plot of residual estimators $\eta_i, 1\leq i \leq 7$ for SIPG and NIPG methods for Model Problem 1.
  • Figure 6.4: Efficiency indices for SIPG and NIPG methods for Model Problem 1.
  • Figure 6.5: Physical setting of Model Problem 2.
  • ...and 3 more figures

Theorems & Definitions (31)

  • Lemma 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Remark 3.1
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 4.1
  • proof
  • Remark 4.2
  • ...and 21 more