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Superconvergent postprocessing of $C^0$ interior penalty method

Ying Cai, Hailong Guo, Zhimin Zhang

TL;DR

Using the argument of superconvergence by difference quotient, it is proved superconvergent results of the recovered Hessian matrix on translation-invariant meshes.

Abstract

This paper focuses on the superconvergence analysis of the Hessian recovery technique for the $C^0$ Interior Penalty Method (C0IP) in solving the biharmonic equation. We establish interior error estimates for C0IP method that serve as the superconvergent analysis tool. Using the argument of superconvergence by difference quotient, we prove superconvergent results of the recovered Hessian matrix on translation-invariant meshes. The Hessian recovery technique enables us to construct an asymptotically exact ${\it a\, posteriori}$ error estimator for the C0IP method. Numerical experiments are provided to support our theoretical results.

Superconvergent postprocessing of $C^0$ interior penalty method

TL;DR

Using the argument of superconvergence by difference quotient, it is proved superconvergent results of the recovered Hessian matrix on translation-invariant meshes.

Abstract

This paper focuses on the superconvergence analysis of the Hessian recovery technique for the Interior Penalty Method (C0IP) in solving the biharmonic equation. We establish interior error estimates for C0IP method that serve as the superconvergent analysis tool. Using the argument of superconvergence by difference quotient, we prove superconvergent results of the recovered Hessian matrix on translation-invariant meshes. The Hessian recovery technique enables us to construct an asymptotically exact error estimator for the C0IP method. Numerical experiments are provided to support our theoretical results.
Paper Structure (14 sections, 26 theorems, 203 equations, 4 figures, 5 tables)

This paper contains 14 sections, 26 theorems, 203 equations, 4 figures, 5 tables.

Table of Contents

  1. Introduction
  2. $C^0$ interior penalty method for biharmonic equations
  3. Biharmonic equations
  4. $C^0$ interior penalty discretization
  5. Hessian recovery method
  6. Interior estimates for C0IP methods
  7. Local error estimates in energy and $L^2$ norms
  8. Interior Maximum Norm Estimates
  9. Superconvergence analysis
  10. Numerical Experiments
  11. Superconvergent results
  12. Adaptive C0IP methods
  13. Conclusion
  14. Some basic estimates and the proof of Lemma \ref{['lem:5.3']}

Key Result

Theorem 3.2

\newlabelthm:pp0 Let $u\in W_{\infty}^{k+2}(\mathcal{K}_z)$ and $u_I=I_hu$; then If $z$ is a node of translation invariant mesh and a mesh symmetric center of the involved nodes and $u\in W_{\infty}^{k+3}(\mathcal{K}_z)$, then Moreover, if $u\in W_{\infty}^{k+4}(\mathcal{K}_z)$ and $k$ is an even number, then

Figures (4)

  • Figure 1: Sampling point selection: (a) quadratic element; (b) cubic element.
  • Figure 1: The loglog plots of errors w.r.t the mesh sizes: (a)regular pattern; (b)Chevron pattern; (c) Criss-cross pattern; (d)Union-Jack pattern.
  • Figure 2: Meshes for Lshape domain problem: (a) Initial mesh; (b) Adaptively refined mesh.
  • Figure 3: Numerical result for test case 3: (a) Numerical errors; (b) Effective index.

Theorems & Definitions (47)

  • Remark 3.1
  • Theorem 3.2
  • Lemma 4.1
  • Proof 1
  • Lemma 4.2
  • Proof 2
  • Lemma 4.3
  • Proof 3
  • Lemma 4.4
  • Proof 4
  • ...and 37 more