arXiv:2610.31822v1 Announce Type: new Abstract: The approximation theory by finite elements is usually described on the simplified geometry of polygonal or polyhedral domains. While avoiding the question of boundary approximation makes the analysis simpler, real applications may involve piecewise smooth domains and in some cases the extension of classical results to general boundaries may be tricky. In this paper we discuss in a systematic way the approximation properties and the error analysis of standard non-isoparametric finite element spaces on piecewise curvilinear domains in two and three dimensions. We show in detail how the geometric error introduces a barrier for the $H^1$ and $L^2$ error of the solution of both Dirichlet and Neumann elliptic problems independently of the regularity of the problem and of the degree of the finite element spaces used.