The minimal angle condition for quadrilateral finite elements of arbitrary degree

TL;DR

This paper establishes that the error bounds for quadrilateral finite elements depend on the minimal interior angle for certain p-values, with sharpness demonstrated through counterexamples.

Contribution

It introduces minimal angle conditions for error estimates in quadrilateral finite elements of arbitrary degree, extending previous understanding to a broader p-range.

Findings

Error constant bounded by minimal interior angle for p=2

Extension of bounds to 1≤p<3

Dependence on maximal angle for p≥3

Abstract

We study $W^{1,p}$ Lagrange interpolation error estimates for general quadrilateral $\mathcal{Q}_{k}$ finite elements with $k\ge 2$. For the most standard case of $p=2$ it turns out that the constant $C$ involved in the error estimate can be bounded in terms of the minimal interior angle of the quadrilateral. Moreover, the same holds for any $p$ in the range $1\le p<3$. On the other hand, for $3\le p$ we show that $C$ also depends on the maximal interior angle. We provide some counterexamples showing that our results are sharp.