Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Rate-optimal higher-order adaptive conforming FEM for biharmonic eigenvalue problems on polygonal domains

Carsten Carstensen, Benedikt Gräßle

TL;DR

This work develops rate-optimal adaptive finite element methods for biharmonic eigenvalue problems on polygonal domains using hierarchical Argyris spaces. It introduces an explicit residual-based a posteriori estimator and proves four adaptivity axioms (stability, reduction, discrete reliability, quasi-orthogonality) to establish optimal convergence toward a simple eigenvalue. The approach is supported by extensive numerical experiments, including high-precision eigenvalues on the unit square and L-shaped domain, isospectral-domain studies under simply-supported BC, and other challenging geometries, demonstrating the practical efficiency and necessity of adaptivity for high-order conforming methods. The results justify the higher computational cost of conforming high-order schemes over low-order nonconforming ones and provide a robust framework for accurate eigenvalue computation on complex polygonal domains.

Abstract

The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long time to realise the necessity of an extension of the classical finite element spaces to make them hierarchical. This paper establishes the novel adaptive schemes for the biharmonic eigenvalue problems and provides a mathematical proof of optimal convergence rates towards a simple eigenvalue and numerical evidence thereof. This makes the suggested algorithm highly competitive and clearly justifies the higher computational and implementational costs compared to low-order nonconforming schemes. The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates and underline that adaptive mesh-refining is mandatory. Five computational benchmarks display accurate reference eigenvalues up to 30 digits.

Rate-optimal higher-order adaptive conforming FEM for biharmonic eigenvalue problems on polygonal domains

TL;DR

This work develops rate-optimal adaptive finite element methods for biharmonic eigenvalue problems on polygonal domains using hierarchical Argyris spaces. It introduces an explicit residual-based a posteriori estimator and proves four adaptivity axioms (stability, reduction, discrete reliability, quasi-orthogonality) to establish optimal convergence toward a simple eigenvalue. The approach is supported by extensive numerical experiments, including high-precision eigenvalues on the unit square and L-shaped domain, isospectral-domain studies under simply-supported BC, and other challenging geometries, demonstrating the practical efficiency and necessity of adaptivity for high-order conforming methods. The results justify the higher computational cost of conforming high-order schemes over low-order nonconforming ones and provide a robust framework for accurate eigenvalue computation on complex polygonal domains.

Abstract

The a posteriori error analysis of the classical Argyris finite element methods dates back to 1996, while the optimal convergence rates of associated adaptive finite element schemes are established only very recently in 2021. It took a long time to realise the necessity of an extension of the classical finite element spaces to make them hierarchical. This paper establishes the novel adaptive schemes for the biharmonic eigenvalue problems and provides a mathematical proof of optimal convergence rates towards a simple eigenvalue and numerical evidence thereof. This makes the suggested algorithm highly competitive and clearly justifies the higher computational and implementational costs compared to low-order nonconforming schemes. The numerical experiments provide overwhelming evidence that higher polynomial degrees pay off with higher convergence rates and underline that adaptive mesh-refining is mandatory. Five computational benchmarks display accurate reference eigenvalues up to 30 digits.
Paper Structure (29 sections, 3 theorems, 90 equations, 9 figures, 5 tables)

This paper contains 29 sections, 3 theorems, 90 equations, 9 figures, 5 tables.

Table of Contents

  1. Introduction
  2. State of the art
  3. Overview
  4. Biharmonic eigenvalue problem and discretisation
  5. Explicit residual-based a posteriori error estimator
  6. Adaptive algorithm AFEM
  7. Outline
  8. General notation
  9. Foundations of a posteriori analysis for conforming EVP
  10. Abstract elliptic eigenvalue problem
  11. Discrete eigenvalue problem
  12. A priori convergence
  13. Two-level notation
  14. Discrete reliability analysis
  15. Argyris FEM for eigenvalue problems
  16. ...and 14 more sections

Key Result

Theorem 1.1

If $\lambda=\lambda_j$ is a simple eigenvalue, then there exists $0<\delta,\Theta$, that exclusively depend on $\Omega$ and the shape-regularity of $\mathbb T$ such that for any $0<s<\infty$ the output $(\mathcal{T}_{\ell})_\ell$ and $(\eta_\ell)_\ell$ of the adaptive algorithm AFEM with bulk parame

Figures (9)

  • Figure 1: Adaptive Argyris finite element scheme AFEM
  • Figure 2: Initial and adapative triangulations of the unit square and the L-shaped domain
  • Figure 3: Convergence history plot of AFEM (hollow) and uniform-red refinement (filled) towards the principal eigenvalue $\lambda_1$ of the unit square for different $p$ and a successive $p$-increase
  • Figure 4: Convergence history plot of $\lambda_1$ and $\lambda_{10}$ from AFEM ($p=5$ left and $p=7$ right) with $n=0,1,2,3$-times red-refinement of $\mathcal{T}_0$ from Figure \ref{['fig:Square_Lshape_mesh']} as initial triangulation
  • Figure 5: Convergence history plot of $\lambda_1$ and $\lambda_{9}$ on the two domains of Figure \ref{['fig:Drum_mesh']} with simply-supported (left) and clamped boundary conditions (right)
  • ...and 4 more figures

Theorems & Definitions (9)

  • Theorem 1.1: optimal rates
  • Lemma 2.1: key
  • proof
  • Lemma 2.2: 2-level a posteriori error estimation
  • proof : Proof of Lemma \ref{['lem:err_Res_est']}.a
  • proof : Proof of Lemma \ref{['lem:err_Res_est']}.b
  • Remark : asymptotic exactness
  • Remark 4.1: Isospectrality for simply-supported boundary conditions
  • Remark 4.2