{"paper":{"title":"Extending Data to Improve Stability and Error Estimates Using Asymmetric Kansa-like Methods to Solve PDEs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"Francis J. Narcowich, Joseph D. Ward, Thomas Hangelbroek","submitted_at":"2025-07-20T22:20:16Z","abstract_excerpt":"In this paper, a theoretical framework is presented\n  for the use of a Kansa-like method to numerically solve elliptic\n  partial differential equations on spheres and other manifolds. The\n  theory addresses both the stability of the method and provides error\n  estimates for two different approximation methods. A Kansa-like\n  matrix is obtained by replacing the test point set $X$, used in the\n  traditional Kansa method, by a larger set $Y$, which is a norming\n  set for the underlying trial space. This gives rise to a rectangular\n  matrix. In addition, if a basis of Lagrange (or local Lagrange)\n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.15137","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/2507.15137/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"}