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Approximation by quadrilateral finite elements

Douglas N. Arnold, Daniele Boffi, Richard S. Falk

TL;DR

The paper analyzes finite element approximation on quadrilateral meshes via bilinear mappings from a reference square, proving that achieving $O(h^{r+1})$ in $L^2$ and $O(h^{r})$ in $H^1$ requires the reference space to contain $\,\mathcal{Q}_r(\\hat{K})$, and that this condition is necessary through a concrete counterexample. It contrasts this with the affine-map case, where containing $\,\mathcal{P}_r(\\hat{K})$ suffices, and shows that quadrilateral meshes can suffer degraded convergence (e.g., for serendipity spaces) unless the stronger condition holds. The paper also discusses asymptotically parallelogram meshes, showing that $\,\mathcal{P}_r(\\hat{K})$ suffices in that special setting, and provides numerical results that illustrate the theoretical degradation and recovery of rates depending on mesh geometry. Overall, the work clarifies when quadrilateral finite elements preserve optimal convergence and explains practical implications for serendipity and mixed/nonconforming elements.

Abstract

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r+1 in L2 and order r in H1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

Approximation by quadrilateral finite elements

TL;DR

The paper analyzes finite element approximation on quadrilateral meshes via bilinear mappings from a reference square, proving that achieving in and in requires the reference space to contain , and that this condition is necessary through a concrete counterexample. It contrasts this with the affine-map case, where containing suffices, and shows that quadrilateral meshes can suffer degraded convergence (e.g., for serendipity spaces) unless the stronger condition holds. The paper also discusses asymptotically parallelogram meshes, showing that suffices in that special setting, and provides numerical results that illustrate the theoretical degradation and recovery of rates depending on mesh geometry. Overall, the work clarifies when quadrilateral finite elements preserve optimal convergence and explains practical implications for serendipity and mixed/nonconforming elements.

Abstract

We consider the approximation properties of finite element spaces on quadrilateral meshes. The finite element spaces are constructed starting with a given finite dimensional space of functions on a square reference element, which is then transformed to a space of functions on each convex quadrilateral element via a bilinear isomorphism of the square onto the element. It is known that for affine isomorphisms, a necessary and sufficient condition for approximation of order r+1 in L2 and order r in H1 is that the given space of functions on the reference element contain all polynomial functions of total degree at most r. In the case of bilinear isomorphisms, it is known that the same estimates hold if the function space contains all polynomial functions of separate degree r. We show, by means of a counterexample, that this latter condition is also necessary. As applications we demonstrate degradation of the convergence order on quadrilateral meshes as compared to rectangular meshes for serendipity finite elements and for various mixed and nonconforming finite elements.

Paper Structure

This paper contains 4 sections, 4 theorems, 24 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Approximation theory of rectangular elements
  3. Approximation theory of quadrilateral elements
  4. Numerical results

Key Result

Theorem 1

Let $\hat{S}$ be a finite dimensional subspace of $L^2(\hat{K})$, $r$ a non-negative integer. The following conditions are equivalent:

Figures (2)

  • Figure 1: a. A partition of the square into four trapezoids. b. A mesh composed of translated dilates of this partition.
  • Figure 2: Three sequences of meshes of the unit square: square, trapezoidal, and asymptotically parallelogram. Each is shown for $n=2$ ,$4$, $8$, and $16$.

Theorems & Definitions (9)

  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • Theorem 4
  • Remark
  • proof