Extenting of Babuška-Aziz's theorem to higher-order Lagrange interpolation
For researchers in finite element methods, this is an incremental extension of a known result to higher-order elements.
The paper extends Babuška-Aziz's theorem to higher-order Lagrange interpolation on triangles and tetrahedrons, showing that squeezing these elements does not degrade the optimal approximation order.
We consider the error analysis of Lagrange interpolation on triangles and tetrahedrons. For Lagrange interpolation of order one, Babuška 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 Babuška-Aziz.