Lower Bounds of the Discretization for Piecewise Polynomials

Assume that $V_h$ is a space of piecewise polynomials of degree less than $r\geq 1$ on a family of quasi-uniform triangulation of size $h$. Then the following well-known upper bound holds for a sufficiently smooth function $u$ and $p\in [1, \infty]$ $$\inf_{v_h\in V_h}\|u-v_h\|_{j,p,\Omega,h} \le C h^{r-j} |u|_{r,p,\Omega},\quad 0\le j\le r.$$ In this paper, we prove that, roughly speaking, if $u\not\in V_h$, the above estimate is sharp. Namely, $$\inf_{v_h\in V_h}\|u-v_h\|_{j,p,\Omega,h} \ge c h^{r-j},\quad 0\le j\le r, \ \ 1\leq p\leq \infty,$$ for some $c>0$. The above result is further extended to various situations including more general Sobolev space norms, 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.