Extenting of Babu\v{s}ka-Aziz's theorem to higher-order Lagrange interpolation

TL;DR

This paper extends Babuška-Aziz's theorem on the stability of linear Lagrange interpolation under geometric squeezing to higher-order interpolations on triangles and tetrahedrons, using difference quotients.

Contribution

The authors generalize the error analysis of Lagrange interpolation to higher orders and higher dimensions, building upon the original Babuška-Aziz technique.

Findings

Error estimates are valid for higher-order interpolations on squeezed elements.

The extension applies to both triangles and tetrahedrons.

The method uses difference quotients for multivariable functions.

Abstract

We consider the error analysis of Lagrange interpolation on triangles and tetrahedrons. For Lagrange interpolation of order one, Babu\v{s}ka and Aziz showed that squeezing a right isosceles triangle perpendicularly does not deteriorate the optimal approximation order. We extend their technique and result to higher-order Lagrange interpolation on both triangles and tetrahedrons. To this end, we make use of difference quotients of functions with two or three variables. Then, the error estimates on squeezed triangles and tetrahedrons are proved by a method that is a straightforward extension of the original given by Babu\v{s}ka-Aziz.