Lower bounds of the discretization error for piecewise polynomials

Assume that V_h is a space of piecewise polynomials of a degree less than r ≥ 1 on a family of quasi-uniform triangulation of size h. There exists the well-known upper bound of the approximation error by V_h for a sufficiently smooth function. In this paper, we prove that, roughly speaking, if the function does not belong to V_h, the upper-bound error estimate is also sharp. This result is further extended to various situations including general shape regular grids and many different types of finite element spaces. As an application, the sharpness of finite element approximation of elliptic problems and the corresponding eigenvalue problems is established.