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Supremum norm A Posteriori Error control of Quadratic Finite Element Method for the Signorini problem

Rohit Khandelwal, Kamana Porwal, Tanvi Wadhawan

TL;DR

This work addresses the 2D Signorini problem by deriving a residual-based a posteriori error estimator in the supremum norm for quadratic conforming FEM. A key contributions are the construction of a discrete contact force density with a sign property, the introduction of a quasi discrete density, and the use of barrier functions and Green's matrix estimates to obtain pointwise reliability and local efficiency. The proposed estimator provides rigorous, computable bounds linking the pointwise errors in the displacement and contact density to a global estimator $\eta_h$, enabling effective adaptive refinement that localizes singularities near the contact boundary. Numerical experiments confirm optimal convergence rates and demonstrate robust performance on adaptive meshes, highlighting the estimator’s practical impact for localized error control in Signorini-type problems.

Abstract

In this paper, we develop a new residual-based pointwise a posteriori error estimator of the quadratic finite element method for the Signorini problem. The supremum norm a posteriori error estimates enable us to locate the singularities locally to control the pointwise errors. In the analysis the discrete counterpart of contact force density is constructed suitably to exhibit the desired sign property. We employ a priori estimates for the standard Green's matrix for the divergence type operator and introduce the upper and lower barriers functions by appropriately modifying the discrete solution. Finally, we present numerical experiments that illustrate the excellent performance of the proposed error estimator.

Supremum norm A Posteriori Error control of Quadratic Finite Element Method for the Signorini problem

TL;DR

This work addresses the 2D Signorini problem by deriving a residual-based a posteriori error estimator in the supremum norm for quadratic conforming FEM. A key contributions are the construction of a discrete contact force density with a sign property, the introduction of a quasi discrete density, and the use of barrier functions and Green's matrix estimates to obtain pointwise reliability and local efficiency. The proposed estimator provides rigorous, computable bounds linking the pointwise errors in the displacement and contact density to a global estimator , enabling effective adaptive refinement that localizes singularities near the contact boundary. Numerical experiments confirm optimal convergence rates and demonstrate robust performance on adaptive meshes, highlighting the estimator’s practical impact for localized error control in Signorini-type problems.

Abstract

In this paper, we develop a new residual-based pointwise a posteriori error estimator of the quadratic finite element method for the Signorini problem. The supremum norm a posteriori error estimates enable us to locate the singularities locally to control the pointwise errors. In the analysis the discrete counterpart of contact force density is constructed suitably to exhibit the desired sign property. We employ a priori estimates for the standard Green's matrix for the divergence type operator and introduce the upper and lower barriers functions by appropriately modifying the discrete solution. Finally, we present numerical experiments that illustrate the excellent performance of the proposed error estimator.
Paper Structure (10 sections, 19 theorems, 150 equations, 4 figures)

This paper contains 10 sections, 19 theorems, 150 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Continuous contact force denisty and Green's matrix
  3. Green's matrix
  4. Preliminaries and discrete variational inequality
  5. Discrete contact force density
  6. quasi discrete contact force density
  7. A posteriori estimates
  8. Reliability of the error estimator
  9. Efficiency of the error estimator
  10. Numerical Results

Key Result

Lemma 2.4

It holds that

Figures (4)

  • Figure 1: Model Problem
  • Figure 2: Plot of error, estimator and efficiency index for Example 7.1
  • Figure 3: Estimator and Adaptive mesh for Example 7.2
  • Figure 4: Plot of individual estimators

Theorems & Definitions (42)

  • Remark 1.1
  • Remark 1.2
  • Definition 2.1
  • Remark 2.2
  • Remark 2.3
  • Lemma 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Remark 3.2
  • Lemma 3.3
  • ...and 32 more