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The Adini finite element on locally refined meshes

Dietmar Gallistl

TL;DR

The paper develops and analyzes a locally refined Adini finite element method for the planar biharmonic equation on 1-irregular rectangular meshes. It proves that quadratic convergence is lost in the presence of hanging nodes unless the normal-derivative DOF is assigned via a uniquely determined averaging rule, which in turn yields a superlinear rate $h^{3/2}$ on uniformly refined meshes; adaptive refinement maintains a first-order behavior. It provides a rigorous a priori error bound and demonstrates the necessity of the averaging assignment by establishing a matching lower bound for any admissible hanging-node coupling. An explicit residual-based a posteriori error estimator is shown to be reliable and efficient (up to best-approximation of the Hessian), and the method extends to various boundary conditions. Numerical experiments confirm the theoretical rates and illustrate the method’s effectiveness for curved and non-rectilinear domains as well as adaptive refinement scenarios.

Abstract

This work introduces a locally refined version of the Adini finite element for the planar biharmonic equation on rectangular partitions with at most one hanging node per edge. If global continuity of the discrete functions is enforced, for such method there is some freedom in assigning the normal derivative degree of freedom at the hanging nodes. It is proven that the convergence order $h^2$ known for regular solutions and regular partitions is lost for any such choice, and that assigning the average of the normal derivatives at the neighbouring regular vertices is the only choice that achieves a superlinear order, namely $h^{3/2}$ on uniformly refined meshes. On adaptive meshes, the method behaves like a first-order scheme. Furthermore, the reliability and efficiency of an explicit residual-based error estimator are shown up to the best approximation of the Hessian by certain piecewise polynomial functions.

The Adini finite element on locally refined meshes

TL;DR

The paper develops and analyzes a locally refined Adini finite element method for the planar biharmonic equation on 1-irregular rectangular meshes. It proves that quadratic convergence is lost in the presence of hanging nodes unless the normal-derivative DOF is assigned via a uniquely determined averaging rule, which in turn yields a superlinear rate on uniformly refined meshes; adaptive refinement maintains a first-order behavior. It provides a rigorous a priori error bound and demonstrates the necessity of the averaging assignment by establishing a matching lower bound for any admissible hanging-node coupling. An explicit residual-based a posteriori error estimator is shown to be reliable and efficient (up to best-approximation of the Hessian), and the method extends to various boundary conditions. Numerical experiments confirm the theoretical rates and illustrate the method’s effectiveness for curved and non-rectilinear domains as well as adaptive refinement scenarios.

Abstract

This work introduces a locally refined version of the Adini finite element for the planar biharmonic equation on rectangular partitions with at most one hanging node per edge. If global continuity of the discrete functions is enforced, for such method there is some freedom in assigning the normal derivative degree of freedom at the hanging nodes. It is proven that the convergence order known for regular solutions and regular partitions is lost for any such choice, and that assigning the average of the normal derivatives at the neighbouring regular vertices is the only choice that achieves a superlinear order, namely on uniformly refined meshes. On adaptive meshes, the method behaves like a first-order scheme. Furthermore, the reliability and efficiency of an explicit residual-based error estimator are shown up to the best approximation of the Hessian by certain piecewise polynomial functions.
Paper Structure (16 sections, 14 theorems, 115 equations, 9 figures)

This paper contains 16 sections, 14 theorems, 115 equations, 9 figures.

Table of Contents

  1. Introduction and main results
  2. Adini's finite element for the biharmonic equation
  3. Continuity conditions at hanging nodes
  4. Analysis of the consistency error, Proof of Theorem \ref{['t:pri']}
  5. A posteriori error estimate, Proof of Theorem \ref{['t:post']}
  6. Extension to other boundary conditions
  7. Numerical results
  8. Illustration of Theorem \ref{['t:pri']} on quasi-uniform meshes
  9. Approximation of a curvilinear domain
  10. Adaptive mesh refinement
  11. Approximation of a non-rectilinear domain
  12. Mixed boundary conditions
  13. Properties of the Adini basis functions
  14. Bilinear interpolation with hanging-node constraint
  15. BFS averaging
  16. ...and 1 more sections

Key Result

Theorem A

Let $f\in L^2(\Omega)$ be such that the exact solution $u$ to the biharmonic problem e:bih satisfies $u\in H^4(\Omega)\cap W^{3,\infty}(\Omega)$. Let $(\mathcal{T}_h)_h$ be a sequence of uniform refinements of an initial partition that satisfies the mesh condition of Definition d:meshcondition and c where $\cup \mathcal{T}^{\mathrm{irr}}$ from e:Tirrdef is the area covered by elements with irregul

Figures (9)

  • Figure 1: Mnemonic diagram of Adini's finite element (left); degrees of freedom at a hanging node (right).
  • Figure 2: Mesh configurations excluded by Definition \ref{['d:meshcondition']}. Left: some neighbouring vertices are irregular. Right: an edge contains more than one irregular vertex.
  • Figure 3: Left: Configuration with a hanging node $\tilde{z}$. Right: Mesh configuration with a regular vertex $z$ and exactly one irregular vertex $\tilde{z}$ on $\partial\omega_z$.
  • Figure 4: Notation for the reference rectangle used in Lemma \ref{['l:admL']}.
  • Figure 5: Left: Numerical illustration of Theorem \ref{['t:pri']} with the errors $\lvert\!\lvert\!\lvert u-u_h\rvert\!\rvert\!\rvert_h$ for a smooth $u$ on the unit square, setting of §\ref{['ss:result_unif']}. Right: Convergence history for the disk domain from §\ref{['ss:results_Circ']}.
  • ...and 4 more figures

Theorems & Definitions (28)

  • Theorem A: a priori error estimate
  • Theorem B: a posteriori error estimate
  • Definition 2.1: mesh condition
  • Definition 3.1: admissible assignment
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • proof
  • Lemma 4.1
  • proof
  • ...and 18 more