{"paper":{"title":"Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Kassem Mustapha, William McLean","submitted_at":"2012-06-12T22:38:17Z","abstract_excerpt":"We consider an initial-boundary value problem for $\\partial_tu-\\partial_t^{-\\alpha}\\nabla^2u=f(t)$, that is, for a fractional diffusion ($-1<\\alpha<0$) or wave ($0<\\alpha<1$) equation. A numerical solution is found by applying a piecewise-linear, discontinuous Galerkin method in time combined with a piecewise-linear, conforming finite element method in space. The time mesh is graded appropriately near $t=0$, but the spatial mesh is quasiuniform. Previously, we proved that the error, measured in the spatial $L_2$-norm, is of order $k^{2+\\alpha_-}+h^2\\ell(k)$, uniformly in $t$, where $k$ is the "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2686","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":""},"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"}