{"paper":{"title":"Superconvergence of the $Q_{k+1,k}$-$Q_{k,k+1}$ divergence-free finite element","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NA","authors_text":"Shangyou Zhang, Yunqing Huang","submitted_at":"2011-12-16T03:30:38Z","abstract_excerpt":"By the standard theory, the stable $Q_{k+1,k}$-$Q_{k,k+1}/Q_{k}^{dc'}$ divergence-free element converges with the optimal order of approximation for the Stokes equations, but only order $k$ for the velocity in $H^1$-norm and the pressure in $L^2$-norm. This is due to one polynomial degree less in $y$ direction for the first component of velocity, a $Q_{k+1,k}$ polynomial. In this manuscript, we will show a superconvergence of the divergence free element that the order of convergence is truly $k+1$, for both velocity and pressure. Numerical tests are provided confirming the sharpness of the the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1112.3705","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"}