{"paper":{"title":"Galerkin Methods for Parabolic and SCHR{\\\"O}DINGER Equations with Dynamical Boundary Conditions and Applications to Underwater Acoustics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.NA"],"primary_cat":"math.NA","authors_text":"D. C. Antonopoulou, G. E. Zouraris, V. A. Dougalis","submitted_at":"2009-04-24T16:11:43Z","abstract_excerpt":"In this paper we consider Galerkin-finite element methods that approximate the solutions of initial-boundary-value problems in one space dimension for parabolic and Schr\\\"odinger evolution equations with dynamical boundary conditions. Error estimates of optimal rates of convergence in $L^2$ and $H^1$ are proved for the accociated semidiscrete and fully discrete Crank-Nicolson-Galerkin approximations. The problem involving the Schr\\\"odinger equation is motivated by considering the standard `parabolic' (paraxial) approximation to the Helmholtz equation, used in underwater acoustics to model long"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0904.3900","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/0904.3900/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"}