{"paper":{"title":"Lower Bounds of the Discretization for Piecewise Polynomials","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Hehu Xie, Jinchao Xu, Qun Lin","submitted_at":"2011-06-22T08:57:19Z","abstract_excerpt":"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, $$\n  \\inf_{v_h\\in V_h}\\|u-v_h\\|_{j,p,\\Omega,h} \\ge c h^{r-j},\\quad 0\\le j\\le r, \\ \\ 1\\leq p\\leq \\infty, $$ for some $c>0$.\n  The above result is fur"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.4395","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}