{"paper":{"title":"Complementary families of approximating polynomials with applications to finite element methods applied to differential equations of arbitrary even spatial order","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Peter K Moore","submitted_at":"2026-06-30T15:36:20Z","abstract_excerpt":"Complementary families of polynomials are introduced to generate $C^m$ finite element basis functions of order $p \\geq 2m+2$ for arbitrary $m \\ge 0$. One family consists of the Hermite splines that serve as the nodal basis functions by ensuring $C^m$ continuity across element boundaries. Explicit formulas for these splines for any $m \\ge 0$ are presented on the canonical interval $[0,1]$. The second family is derived on the interval $[-1,1]$ from derivatives of order $m+1$ of the Legendre polynomials of degree $p-m-1$ multiplied by binomial powers of degree $m+1$ at -1 and 1, respectively. The"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.31829","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/2606.31829/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"}