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A posteriori error analysis of hybrid higher order methods for the elliptic obstacle problem

Kamana Porwal, Ritesh Singla

TL;DR

This work addresses reliable a posteriori error control for the elliptic obstacle problem using a Hybrid High-Order (HHO) discretization with cell unknowns of degree $r=0$ and face unknowns of degree $s\in\{0,1\}$, where discrete obstacle constraints are applied in cells. The authors construct a residual-based estimator from a discrete Lagrange multiplier and a linear averaging operator, and prove both reliability and efficiency of the estimator in the energy norm and the $H^{-1}$ norm, respectively, supported by 2D numerical experiments. The methodology hinges on a gradient reconstruction operator, an averaging map, and a carefully defined residual $\mathfrak{R}_h$, enabling adaptive refinement that captures both smooth regions and singular features. The results demonstrate optimal convergence rates with respect to degrees of freedom and validate the practical viability of HHO methods for variational inequalities like the obstacle problem.

Abstract

In this article, a posteriori error analysis of the elliptic obstacle problem is addressed using hybrid high-order methods. The method involve cell unknowns represented by degree-$r$ polynomials and face unknowns represented by degree-$s$ polynomials, where $r=0$ and $s$ is either $0$ or $1$. The discrete obstacle constraints are specifically applied to the cell unknowns. The analysis hinges on the construction of a suitable Lagrange multiplier, a residual functional and a linear averaging map. The reliability and the efficiency of the proposed a posteriori error estimator is discussed, and the study is concluded by numerical experiments supporting the theoretical results.

A posteriori error analysis of hybrid higher order methods for the elliptic obstacle problem

TL;DR

This work addresses reliable a posteriori error control for the elliptic obstacle problem using a Hybrid High-Order (HHO) discretization with cell unknowns of degree and face unknowns of degree , where discrete obstacle constraints are applied in cells. The authors construct a residual-based estimator from a discrete Lagrange multiplier and a linear averaging operator, and prove both reliability and efficiency of the estimator in the energy norm and the norm, respectively, supported by 2D numerical experiments. The methodology hinges on a gradient reconstruction operator, an averaging map, and a carefully defined residual , enabling adaptive refinement that captures both smooth regions and singular features. The results demonstrate optimal convergence rates with respect to degrees of freedom and validate the practical viability of HHO methods for variational inequalities like the obstacle problem.

Abstract

In this article, a posteriori error analysis of the elliptic obstacle problem is addressed using hybrid high-order methods. The method involve cell unknowns represented by degree- polynomials and face unknowns represented by degree- polynomials, where and is either or . The discrete obstacle constraints are specifically applied to the cell unknowns. The analysis hinges on the construction of a suitable Lagrange multiplier, a residual functional and a linear averaging map. The reliability and the efficiency of the proposed a posteriori error estimator is discussed, and the study is concluded by numerical experiments supporting the theoretical results.
Paper Structure (11 sections, 9 theorems, 85 equations, 3 figures)

This paper contains 11 sections, 9 theorems, 85 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Hybrid High-Order Discretization
  3. Discrete Setting
  4. Degrees of Freedom
  5. Local Gradient Reconstruction Operator
  6. Discrete Problem
  7. Averaging Operator
  8. A Posteriori Error Analysis
  9. Reliability of the Error Estimator
  10. Efficiency of the Error Estimator
  11. Numerical Experiments

Key Result

Lemma 2.1

Let $T\in\mathcal{T}_h$ and let $F\in\mathcal{F}_T$ denote an edge/face of the simplex $T$. The following relations hold where $\mathbb P^l(T)$ represents the space of all $d$-variate polynomial function of degree atmost $l$.

Figures (3)

  • Figure 4.1: Error, Estimator and Efficiency Index for Example 1.
  • Figure 4.2: Adaptive Mesh for Example 1 and Example 2 respectively.
  • Figure 4.3: Error, Estimator and Efficiency Index for Example 2.

Theorems & Definitions (22)

  • Remark 1.1
  • Lemma 2.1: Trace Inequality bren
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Lemma 2.2
  • proof
  • Remark 2.5
  • Remark 2.6
  • ...and 12 more