Some results of anisotropic interpolation of function in $W^{2,p}(\Omega)$ Sobolev Spaces with explicit constants

In this paper, we give some general results of $P_1$ interpolation error of function in $W^{2,p}(\Omega)$ Sobolev Spaces, on sequences of conformal, possibly anisotropic, meshes for spaces of dimension $d=2$ or $d=3$. The upper and lower bounds of the interpolation error are given with explicit constants. The idea is to define a metric such that the volume and perimeter of a simplex measured in this metric controls the error standard. The goal is to use the Hessian matrix to search for a quasi-optimal mesh, or for a posteriori error estimation and mesh adaptation strategy for the resolution of PDE problems by finite element method.