Anisotropic error bounds of Lagrange interpolation with any order in two and three dimensions

In this paper, using the Newton's formula of Lagrange interpolation, we present a new proof of the anisotropic error bounds for Lagrange interpolation of any order on the triangle, rectangle, tetrahedron and cube in a unified way. It is known that the polynomial interpolations are the foundations of construction the finite elements and the interpolation error estimates play a key role in deriving a priori error estimates of the finite element methods. The main strategy of the traditional interpolation theory is fairly standard, namely, first deriving the estimate on the reference element and then an application of a coordinate transformation between a general element and the reference element, see [11,7] and references therein. For the triangular and rectangular elements in two dimension and the tetrahedral and cubic elements in three dimension, the mapping between a general element and the reference element is an affine mapping, so in the following we call these elements affine elements. The classical error estimates of the polynomial interpolation on the affine elements need the regular [11] or nondegenerate [7] condition, i.e., the ratio of the diameters of the element and the biggest ball contained in the element is uniformly bounded. This condition restricts the applications of the finite elements. It is found (see e.g., [6,15]) a long time ago that this condition is not necessary for some interpolation error estimates. We call the element does not satisfy the regular condition the anisotropic element. Recently, the research of the anisotropic elements is rapidly developed , and there are several different methods dealing with them. Apel and Dobrowolski [3], Apel [4] gave one anisotropic form of the interpolation error on the reference element. They got the anisotropic interpolation error estimates on a general element for some Lagrange and Hermite elements under the maximal angle and coordinate system conditions. The corresponding appeared derivatives are along the coordinate directions. Chen et al. [9,10] extend this method by presenting a simple anisotropic criterion on the reference element and analyzed some nonconforming elements. Acosta [1], Acosta and Duran [2], Duran [12,13] got the anisotropic error estimates for low order Lagrange and R-T interpolations by using of the average property of the interpolation and the appeared derivatives under consideration are along the directions of the element boundary. The different forms of the anisotropic error estimate of the linear triangular Lagrange interpolation are obtained by the decomposition of the transformation matrix between a …