Let us call a {\bf highly heterogeneous function} a function that is either locally singular or a smooth function but, with too small details in comparison with domain size. We study the $L^2$ norm of the interpolation error $E_h$ between a function $u$ and $\Pi_h$ $u$ its $P1$ continuous interpolate: we use four examples of functions, that represent different cases of {\bf highly heterogeneous functions}. Comparing first the convergence of $E_h$ as a function of number of nodes on uniform or adaptive meshes, we observe a convergence of order 2, only for a smooth function when the number of nodes is sufficiently large, when an uniform sequence of mesh is choosen. Conversely, almost always holds second-order convergence when an adaptive mesh algorithm is applied. We give some theoretical arguments concerning this phenomenon. Following some ideas currently used in spectral methods, we consider the $P1$ approximation of $u$ on nested meshes and express the representation of $u_h$ as a {\bf series} with increasing fineness of its terms. The size of each terms as a function of the corresponding level number is examined.