Lower Bounds of the Discretization for Piecewise Polynomials ∗

@inproceedings{Lin2011LowerBO,
  title={Lower Bounds of the Discretization for Piecewise Polynomials ∗},
  author={Qun Lin and Hehu Xie and Jinchao Xu},
  year={2011}
}
Assume that Vh is a space of piecewise polynomials of degree less than r ≥ 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 ∈ [1,∞] inf vh∈Vh ‖u− vh‖j,p,Ω,h ≤ Ch |u|r,p,Ω, 0 ≤ j ≤ r. In this paper, we prove that, roughly speaking, if u 6∈ Vh, the above estimate is sharp. Namely, inf vh∈Vh ‖u− vh‖j,p,Ω,h ≥ ch r−j , 0 ≤ j ≤ r, 1 ≤ p ≤ ∞, for some c > 0. The above result is further extended to various… CONTINUE READING