An interpolation error estimate in $\mathcal{R}^2$ based on the anisotropic measures of higher order derivatives

In this paper, we introduce the magnitude, orientation, and anisotropic ratio for the higher order derivative $\nabla^{k+1}u$ (with $k\ge 1$) of a function $u$ to characterize its anisotropic behavior. The magnitude is equivalent to its usual Euclidean norm. The orientation is the direction along which the absolute value of the $k+1$-th directional derivative is about the smallest, while along its perpendicular direction it is about the largest. The anisotropic ratio measures the strength of the anisotropic behavior of $\nabla^{k+1}u$. These quantities are invariant under translation and rotation of the independent variables. They correspond to the area, orientation, and aspect ratio for triangular elements. Based on these measures, we derive an anisotropic error estimate for the piecewise polynomial interpolation over a family of triangulations that are quasi-uniform under a given Riemannian metric. Among the meshes of a fixed number of elements it is identified that the interpolation error is nearly the minimum on the one in which all the elements are aligned with the orientation of $\nabla^{k+1}u$, their aspect ratios are about the anisotropic ratio of $\nabla^{k+1}u$, and their areas make the error evenly distributed over every element.