{"paper":{"title":"Discontinuous Galerkin finite element differential calculus and applications to numerical solutions of linear and nonlinear partial differential equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Michael Neilan, Thomas Lewis, Xiaobing Feng","submitted_at":"2013-02-27T20:35:45Z","abstract_excerpt":"This paper develops a discontinuous Galerkin (DG) finite element differential calculus theory for approximating weak derivatives of Sobolev functions and piecewise Sobolev functions. By introducing numerical one-sided derivatives as building blocks, various first and second order numericaloperators such as the gradient, divergence, Hessian, and Laplacian operator are defined, and their corresponding calculus rules are established. Among the calculus rules are product and chain rules, integration by parts formulas and the divergence theorem. Approximation properties and the relationship between"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.6984","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"}