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A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators

Dietmar Gallistl, Shudan Tian

TL;DR

The error estimator is shown to be reliable and locally efficient up to this polynomial best-approximation error and oscillations of the right-hand side.

Abstract

For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local best-approximation error of the finite element function by piecewise polynomial functions of the degree determining the expected approximation order, which need not coincide with the maximal polynomial degree of the element, for example if bubble functions are used. The error estimator is shown to be reliable and locally efficient up to this polynomial best-approximation error and oscillations of the right-hand side.

A posteriori error estimates for nonconforming discretizations of singularly perturbed biharmonic operators

TL;DR

The error estimator is shown to be reliable and locally efficient up to this polynomial best-approximation error and oscillations of the right-hand side.

Abstract

For the pure biharmonic equation and a biharmonic singular perturbation problem, a residual-based error estimator is introduced which applies to many existing nonconforming finite elements. The error estimator involves the local best-approximation error of the finite element function by piecewise polynomial functions of the degree determining the expected approximation order, which need not coincide with the maximal polynomial degree of the element, for example if bubble functions are used. The error estimator is shown to be reliable and locally efficient up to this polynomial best-approximation error and oscillations of the right-hand side.
Paper Structure (20 sections, 8 theorems, 69 equations, 6 figures, 1 table)

This paper contains 20 sections, 8 theorems, 69 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Weak formulation of the model problem
  4. Discrete functions
  5. Finite element discretization
  6. A posteriori error estimates
  7. Preparatory identities
  8. Error estimator contributions
  9. Error estimator reliability
  10. Error estimator efficiency
  11. Some Examples
  12. Numerical experiments
  13. Setup
  14. Example 1: The biharmonic equation
  15. Example 2: Singularly perturbed problem on the square
  16. ...and 5 more sections

Key Result

lemma 1

The weak continuity e:weak_cont and Assumption assumption:main imply that $a_h$ is positive definite over $V+V_h$.

Figures (6)

  • Figure 1: Convergence history for Example 1.
  • Figure 2: Convergence history for Example 2.
  • Figure 3: Values of the error and estimator for Example 3 with the NTW element.
  • Figure 4: Values of the error and estimator for Example 3 with the MS element.
  • Figure 5: Adaptive mesh with 120 577 (left) and 130,617 (right) degrees of freedom generated by MS element when $\varepsilon=1/100$ and $\varepsilon=0$, respectively, in Example 3.
  • ...and 1 more figures

Theorems & Definitions (19)

  • remark 1
  • remark 2
  • lemma 1
  • proof
  • lemma 2
  • lemma 3: integration by parts
  • proof
  • lemma 4: equilibrium residual
  • proof
  • lemma 5: localization
  • ...and 9 more