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A posteriori error estimates for a modified Morley FEM

A. K. Dond, D. Gallistl, S. Nayak, M. Schedensack

Abstract

Residual-based a~posteriori error estimators are derived for the modified Morley FEM, proposed by Wang, Xu, Hu [J. Comput. Math, 24(2), 2006], for the singularly perturbed biharmonic equation and the nonlinear von Kármán equations. The error estimators are proven to be reliable and efficient. Moreover, an adaptive algorithm driven by these error estimators is investigated in numerical experiments.

A posteriori error estimates for a modified Morley FEM

Abstract

Residual-based a~posteriori error estimators are derived for the modified Morley FEM, proposed by Wang, Xu, Hu [J. Comput. Math, 24(2), 2006], for the singularly perturbed biharmonic equation and the nonlinear von Kármán equations. The error estimators are proven to be reliable and efficient. Moreover, an adaptive algorithm driven by these error estimators is investigated in numerical experiments.
Paper Structure (15 sections, 9 theorems, 165 equations, 3 figures)

This paper contains 15 sections, 9 theorems, 165 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Morley interpolation operator.
  4. Quasi-interpolation operator.
  5. Nodal interpolation operator.
  6. Enriching operator.
  7. Singularly perturbed biharmonic problem
  8. The von Kármán equations
  9. The continuous problem
  10. The discretization and a priori error analysis
  11. Error estimator
  12. Numerical experiments
  13. Singularly perturbed biharmonic equation
  14. Von Kármán equation
  15. Outline of the proof of Theorem \ref{['t:vK_apriori']}

Key Result

Theorem 3.1

The exact solution $u\in H^2_0(\Omega)$ to e:sing_pert_weak_form and the discrete solution $u_h\in \mathcal{M}_0(\mathcal{T})$ to e:sing_pert_discrete_problem satisfy

Figures (3)

  • Figure 1: Adaptive meshes, errors, and estimators in Example \ref{['eg_xBL']}. "Ada" indicates adaptive, "Uni" indicates uniform mesh refinement.
  • Figure 2: Estimators and adaptive mesh with various values of $\varepsilon$ in Example \ref{['eg_LBL']}
  • Figure 3: Comparison of errors and the complete a posteriori estimator for Examples \ref{['eg_vke1']} and \ref{['eg_vke2']}.

Theorems & Definitions (27)

  • Theorem 3.1: reliability
  • Lemma 3.2: a posteriori estimate for nonconformity error
  • proof
  • Remark 3.3
  • Remark 3.4
  • proof : Proof of Theorem \ref{['t:sp_reliability']}
  • Theorem 3.5: efficiency
  • proof
  • Theorem 4.1
  • proof
  • ...and 17 more