Interpolation Error Estimates for Two Dimensions

Abstract

We formulate the fundamental theoretical results which are later employed for the anisotropic mesh adaptation method. First, we recall the geometry terms of a mesh triangle K discussed in the previous chapter. Further, we define an interpolation of a sufficiently smooth function u on element K as a polynomial function having the same value and partial derivatives as the original function at the barycenter of K. Moreover, we derive estimates of the difference between u and its interpolation (=interpolation error estimates) in several norms. These estimates take into account the geometry of mesh element K. Finally, we derive the optimal shape of a triangle with given barycenter, minimizing the interpolation error estimates.