{"paper":{"title":"A sharp error estimate of piecewise polynomial collocation for nonlocal problems with weakly singular kernels","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Jiankang Shi, Jiming Wu, Minghua Chen, Wenya Qi","submitted_at":"2019-09-24T08:17:43Z","abstract_excerpt":"As is well known, using piecewise linear polynomial collocation (PLC) and piecewise quadratic polynomial collocation (PQC), respectively, to approximate the weakly singular integral $$I(a,b,x) =\\int^b_a \\frac{u(y)}{|x-y|^\\gamma}dy, \\quad x \\in (a,b) ,\\quad 0< \\gamma <1,$$ have the local truncation error $\\mathcal{O}\\left(h^2\\right)$ and $\\mathcal{O}\\left(h^{4-\\gamma}\\right)$. Moreover, for Fredholm weakly singular integral equations of the second kind, i.e., $\\lambda u(x)- I(a,b,x) =f(x)$ with $ \\lambda \\neq 0$, also have global convergence rate $\\mathcal{O}\\left(h^2\\right)$ and $\\mathcal{O}\\l"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1909.10756","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1909.10756/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}