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Upper bound of high-order derivatives for Wachspress coordinates on polytopes

Pengjie Tian, Yanqiu Wang

Abstract

The gradient bounds of generalized barycentric coordinates play an essential role in the $H^1$ norm approximation error estimate of generalized barycentric interpolations. Similarly, the $H^k$ norm, $k>1$, estimate needs upper bounds of high-order derivatives, which are not available in the literature. In this paper, we derive such upper bounds for the Wachspress generalized barycentric coordinates on simple convex $d$-dimensional polytopes, $d\ge 1$. The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of, for example, fourth-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.

Upper bound of high-order derivatives for Wachspress coordinates on polytopes

Abstract

The gradient bounds of generalized barycentric coordinates play an essential role in the norm approximation error estimate of generalized barycentric interpolations. Similarly, the norm, , estimate needs upper bounds of high-order derivatives, which are not available in the literature. In this paper, we derive such upper bounds for the Wachspress generalized barycentric coordinates on simple convex -dimensional polytopes, . The result can be used to prove optimal convergence for Wachspress-based polytopal finite element approximation of, for example, fourth-order elliptic equations. Another contribution of this paper is to compare various shape-regularity conditions for simple convex polytopes, and to clarify their relations using knowledge from convex geometry.

Paper Structure

This paper contains 15 sections, 13 theorems, 69 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Wachspress coordinates and some preliminary tools
  3. Convex polytope
  4. Wachspress coordinates
  5. Preliminary tools
  6. Derivatives of Wachspress coordinates and the upper bound
  7. Upper bound of $|D^{\boldsymbol{\nu}}w_{\boldsymbol{v}}|$ and $|D^{\boldsymbol{\nu}}W|$
  8. Lower bound of $W$
  9. Main result: the upper bound of $|D^{\boldsymbol{\alpha}}\phi_{\boldsymbol{v}}|$
  10. Geometric assumptions
  11. Numerical results
  12. Can one easily obtain high quality polygonal meshes with $h_* = O(h_K)$?
  13. When $h_* = O(h_K)$, do numerical results agree well with Theorem \ref{['cor:bound']}
  14. What happens when $h_* \ll h_K$?
  15. When does one need a higher ($|\boldsymbol{\alpha}|>1$) derivative bound?

Key Result

Lemma 2.1

\newlabellem:Wnonzero Function $W$ is positive in $K$.

Figures (24)

  • Figure 3.1: A counter example to the proof of Prop. 8 in GilletteRandBajaj2012.
  • Figure 4.1: A convex polygon is located between two parallel lines (supporting hyperplanes). The minimum distance between all pairs of parallel supporting hyperplanes is the width of the polygon.
  • Figure 4.2: The left polygon does not satisfy Assumption (H5) or (H5'). The right polygon satisfies Assumptions (H5) and (H5'), but not (H1).
  • Figure 5.1: Statistics of polygons in a CVT mesh after eliminating "short" edges. (1) A CVT mesh after eliminating "short" edges. (2)-(3) The distributions of $h_*/h_K$ and $n$-gons in a CVT mesh containing $10000$ polygons. (4) $\frac{h_*}{h_K}$ vs. $n$ for the same CVT mesh.
  • Figure 5.2: Statistics of random convex polygons. (1)-(2) The distributions of $h_*/h_K$ and $n$-gons in $10000$ random convex polygons. (3) $\frac{h_*}{h_K}$ vs. $n$ for the same $10000$ random convex polygons.
  • ...and 19 more figures

Theorems & Definitions (35)

  • Remark 2.1
  • Remark 2.2
  • Lemma 2.1
  • proof
  • Remark 2.3
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Lemma 2.2
  • proof
  • ...and 25 more