{"paper":{"title":"Uniform shift estimates for transmission problems and optimal rates of convergence for the parametric Finite Element Method","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP"],"primary_cat":"math.NA","authors_text":"Hengguang Li, Victor Nistor, Yu Qiao","submitted_at":"2012-12-27T01:35:48Z","abstract_excerpt":"Let $\\Omega \\subset \\RR^d$, $d \\geqslant 1$, be a bounded domain with piecewise smooth boundary $\\partial \\Omega $ and let $U$ be an open subset of a Banach space $Y$. Motivated by questions in \"Uncertainty Quantification,\" we consider a parametric family $P = (P_y)_{y \\in U}$ of uniformly strongly elliptic, second order partial differential operators $P_y$ on $\\Omega$. We allow jump discontinuities in the coefficients. We establish a regularity result for the solution $u: \\Omega \\times U \\to \\RR$ of the parametric, elliptic boundary value/transmission problem $P_y u_y = f_y$, $y \\in U$, with "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.6287","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"}