The minimal angle condition for quadrilateral finite elements of arbitrary degree
Provides precise geometric conditions for error bounds in quadrilateral finite elements, benefiting numerical analysts and engineers using these elements.
The paper establishes that for quadrilateral Q_k finite elements with k≥2, the constant in W^{1,p} Lagrange interpolation error estimates depends only on the minimal interior angle for 1≤p<3, but also on the maximal interior angle for p≥3, with sharpness demonstrated via counterexamples.
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.