{"paper":{"title":"On the relation between Galerkin approximations and canonical best-approximations of solutions to some non-linear Schr\\\"odinger equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Muhammad Hassan, Yipeng Wang, Yvon Maday","submitted_at":"2025-02-11T15:25:13Z","abstract_excerpt":"In this paper, we establish a superconvergence property of Galerkin approximations to some non-linear Schr\\\"odinger equations of Gross-Pitaevskii type. More precisely, denoting by $u^*\\in X \\subseteq H^1(\\Omega)$ the exact solution to such an equation, by $\\{X_{\\delta}\\}_{\\delta >0}$, a sequence of conforming subspaces of $X$ satisfying the approximation property, by $u_\\delta^*\\in X_{\\delta}$ the Galerkin solution to the equation, and by $\\Pi^X_{\\delta} u^*$, the $(\\cdot, \\cdot)_{X}$-best approximation in $X_\\delta$ of $u^*$, we show -- under some assumptions -- that $u_\\delta^*$ converges at"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.07638","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/2502.07638/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"}