Anisotropic Measures of Third Order Derivatives and the Quadratic Interpolation Error on Triangular Elements

The main purpose of this paper is to present a closer look at how the $H^1$- and $L^2$-errors for quadratic interpolation on a triangle are determined by the triangle geometry and the anisotropic behavior of the third order derivatives of interpolated functions. We characterize quantitatively the anisotropic behavior of a third order derivative tensor by its orientation and anisotropic ratio. Both exact error formulas and numerical experiments are presented for model problems of interpolating a cubic function $u$ at the vertices and the midpoints of three sides of a triangle. Based on the study on model problems, we conclude that when an element is aligned with the orientation of $\nabla^3 u$, the aspect ratio leading to nearly the smallest $H^1$- and $L^2$-norms of the interpolation error is approximately equal to the anisotropic ratio of $\nabla^3 u$. With this alignment and aspect ratio taken, the $H^1$-seminorm of the error is proportional to the reciprocal of the anisotropic ratio of $\nabla^3 u$, the $L^2$-norm of the error is proportional to the $-\frac 32$th power of the anisotropic ratio, and both of them are insensitive to the internal angles of the element.