Anisotropic interpolation error estimate for arbitrary quadrilateral isoparametric elements

TL;DR

This paper establishes anisotropic interpolation error estimates for quadrilateral isoparametric elements under simple geometric conditions, improving upon previous results especially for perturbed rectangles.

Contribution

It provides a new, more general error estimate applicable to a wide class of quadrilaterals with simpler geometric conditions than prior methods.

Findings

Error estimates hold for all $p \\in [1,\\infty)$.

Results apply to quadrilaterals satisfying MAC and $clos$ conditions.

Improved estimates for perturbed rectangles compared to previous work.

Abstract

The aim of this paper is to show that, for any $p \in [1,\infty)$, the $W^{1,p}$-anisotropic interpolation error estimate holds on quadrilateral isoparametric elements verifying the maximum angle condition ($MAC$) and the property of comparable lengths for opposite sides ($clos$), i.e., on all those quadrilaterals with interior angles uniformly bounded away from $\pi$ and with both pairs of opposite sides having comparable lengths. For rectangular elements our interpolation error estimate agrees with the usual one whereas for perturbations of rectangles (the most general quadrilateral elements previously considered as far as we know) our result has some advantages with respect to the pre-existing ones: the interpolation error estimate that we proved is written by using two neighboring sides of the element instead of the sides of the unknown perturbed rectangle and, on the other hand, conditions $MAC$ and $clos$ are much simpler requirements to verify and with a clearer geometrical sense than those involved in the definition of perturbations of a rectangle.