Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

5
The categories of this paper are: gr-qc
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
25
The categories of this paper are: patt-sol
8
The categories of this paper are: hep-ex
24
The categories of this paper are: dg-ga
14
The categories of this paper are: math-ph
27
The categories of this paper are: comp-gas
0
The categories of this paper are: hep-ph
37
The categories of this paper are: supr-con
28
The categories of this paper are: alg-geom
29
The categories of this paper are: funct-an
32
The categories of this paper are: atom-ph
3
The categories of this paper are: physics
19
The categories of this paper are: eess