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A posteriori error estimation for weak Galerkin method of the fourth-order singularly perturbed problem

Shicheng Liu, Qilong Zhai

TL;DR

The paper develops a residual-based a posteriori error estimator for the Weak Galerkin discretization of a fourth-order singularly perturbed problem $\varepsilon^{2}\Delta^{2}u-\Delta u=f$, establishing reliability and efficiency. A recovery operator $E$ maps interior WG functions to a $C^1$-conforming space on macro elements, enabling a rigorous upper bound $|||u-u_h|||\le C\eta_h$. Efficiency bounds are obtained via bubble-function arguments, yielding $\eta_h\le C\big(|||u-u_h|||+\text{data oscillation}\big)$. Numerical experiments corroborate the theoretical results, showing that the adaptive WG method with the proposed estimator accurately captures internal and boundary layers and achieves effective mesh refinement. The work provides a robust, computable framework for error control and adaptive refinement in high-order singularly perturbed PDEs using WG discretizations.

Abstract

In this paper, we present a posteriori error estimation for weak Galerkin method applied to fourth order singularly perturbed problem. The weak Galerkin discretization space and numerical scheme are first described. A fully computable residual type error estimator is then constructed. Both the reliability and efficiency of the proposed estimator are rigorously demonstrated. Numerical experiments are provided to validate the theoretical findings.

A posteriori error estimation for weak Galerkin method of the fourth-order singularly perturbed problem

TL;DR

The paper develops a residual-based a posteriori error estimator for the Weak Galerkin discretization of a fourth-order singularly perturbed problem , establishing reliability and efficiency. A recovery operator maps interior WG functions to a -conforming space on macro elements, enabling a rigorous upper bound . Efficiency bounds are obtained via bubble-function arguments, yielding . Numerical experiments corroborate the theoretical results, showing that the adaptive WG method with the proposed estimator accurately captures internal and boundary layers and achieves effective mesh refinement. The work provides a robust, computable framework for error control and adaptive refinement in high-order singularly perturbed PDEs using WG discretizations.

Abstract

In this paper, we present a posteriori error estimation for weak Galerkin method applied to fourth order singularly perturbed problem. The weak Galerkin discretization space and numerical scheme are first described. A fully computable residual type error estimator is then constructed. Both the reliability and efficiency of the proposed estimator are rigorously demonstrated. Numerical experiments are provided to validate the theoretical findings.

Paper Structure

This paper contains 7 sections, 9 theorems, 88 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. Numerical algorithm
  4. A posteriori error estimation
  5. Upper bound
  6. Lower bound
  7. Numerical Experiments

Key Result

Lemma 2.1

On each cell $T \in \mathcal{T}_{h}$, the following identity holds for all $v \in H^2(T)$,

Figures (4)

  • Figure 1: (a) Convergence rates of the error and the error estimator; (b) The final adapted mesh; (c) Exact solution; (d) Numerical solution.
  • Figure 2: (a) Convergence rates of the error and the error estimator; (b) The final adapted mesh; (c) Exact solution; (d) Numerical solution.
  • Figure 3: (a) Convergence rates of the error and the error estimator; (b) The final adapted mesh; (c) Exact solution; (d) Numerical solution.
  • Figure 4: (a) Convergence rates of the error and the error estimator; (b) The final adapted mesh; (c) Numerical solution.

Theorems & Definitions (21)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • proof
  • Lemma 4.1
  • Lemma 4.2
  • proof
  • Lemma 4.3
  • ...and 11 more