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Enhanced gradient recovery-based a posteriori error estimator and adaptive finite element method for elliptic equations

Ying Liu, Jingjing Xiao, Nianyu Yi, Huihui Cao

TL;DR

The paper addresses the lack of guaranteed bounds for gradient-recovery-based a posteriori error estimators in elliptic PDEs. It introduces an enhanced estimator that combines the gradient-recovery discrepancy $||G(\nabla u_h)-\nabla u_h||$ with a residual-like term $||h(f+\nabla\cdot G(\nabla u_h))||$, and proves global reliability and local efficiency. An adaptive algorithm using this estimator and the newest-vertex bisection refinement with tailored marking is shown to converge via a contraction argument. Numerical experiments on challenging problems with interfaces, layered coefficients, and 3D domains demonstrate asymptotic exactness of the estimator and substantial practical efficiency in driving mesh refinement.

Abstract

Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators, one can not establish upper and lower a posteriori error bounds for the classical recovery type error estimators without the saturation assumption. In this paper, we first present three examples to show the unsatisfactory performance in the practice of standard residual or recovery-type error estimators, then, an improved gradient recovery-based a posteriori error estimator is constructed. The proposed error estimator contains two parts, one is the difference between the direct and post-processed gradient approximations, and the other is the residual of the recovered gradient. The reliability and efficiency of the enhanced estimator are derived. Based on the improved recovery-based error estimator and the newest-vertex bisection refinement method with a tailored mark strategy, an adaptive finite element algorithm is designed. We then prove the convergence of the adaptive method by establishing the contraction of gradient error plus oscillation. Numerical experiments are provided to illustrate the asymptotic exactness of the new recovery-based a posteriori error estimator and the high efficiency of the corresponding adaptive algorithm.

Enhanced gradient recovery-based a posteriori error estimator and adaptive finite element method for elliptic equations

TL;DR

The paper addresses the lack of guaranteed bounds for gradient-recovery-based a posteriori error estimators in elliptic PDEs. It introduces an enhanced estimator that combines the gradient-recovery discrepancy with a residual-like term , and proves global reliability and local efficiency. An adaptive algorithm using this estimator and the newest-vertex bisection refinement with tailored marking is shown to converge via a contraction argument. Numerical experiments on challenging problems with interfaces, layered coefficients, and 3D domains demonstrate asymptotic exactness of the estimator and substantial practical efficiency in driving mesh refinement.

Abstract

Recovery type a posteriori error estimators are popular, particularly in the engineering community, for their computationally inexpensive, easy to implement, and generally asymptotically exactness. Unlike the residual type error estimators, one can not establish upper and lower a posteriori error bounds for the classical recovery type error estimators without the saturation assumption. In this paper, we first present three examples to show the unsatisfactory performance in the practice of standard residual or recovery-type error estimators, then, an improved gradient recovery-based a posteriori error estimator is constructed. The proposed error estimator contains two parts, one is the difference between the direct and post-processed gradient approximations, and the other is the residual of the recovered gradient. The reliability and efficiency of the enhanced estimator are derived. Based on the improved recovery-based error estimator and the newest-vertex bisection refinement method with a tailored mark strategy, an adaptive finite element algorithm is designed. We then prove the convergence of the adaptive method by establishing the contraction of gradient error plus oscillation. Numerical experiments are provided to illustrate the asymptotic exactness of the new recovery-based a posteriori error estimator and the high efficiency of the corresponding adaptive algorithm.

Paper Structure

This paper contains 7 sections, 8 theorems, 63 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Model problem and its finite element scheme
  3. Improved gradient recovery-based a posteriori error estimation
  4. Poor performances of classical error estimators
  5. Improved error estimator based on gradient recovery
  6. Adaptive finite element method
  7. Numerical experiments

Key Result

Lemma 3.1

For any $K\in \mathcal{T}_h$, $v\in H_{0,D}^1(\Omega)$, $\omega_K\equiv\cup_{\bar{K'}\cap\bar{K}\neq\emptyset}{K'}$, there exists a positive constant $C$ such that

Figures (10)

  • Figure 1: Local refined mesh and the history of the error and estimator.
  • Figure 2: Distribution of $f$ and mesh
  • Figure 3: Numerical results of adaptive algorithm for Example \ref{['exam3_2']}.
  • Figure 4: Numerical results of Example \ref{['exam3_3']}.
  • Figure 5: Numerical results of Example \ref{['test2']} with error estimator \ref{['estomator']}.
  • ...and 5 more figures

Theorems & Definitions (23)

  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Lemma 3.1: verfurth1999
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Remark 3.1
  • Lemma 4.1
  • ...and 13 more