{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,8,21]],"date-time":"2026-08-21T18:33:08Z","timestamp":1787337188280,"version":"build-2736575974"},"reference-count":35,"publisher":"Society for Industrial & Applied Mathematics (SIAM)","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["SIAM J. Numer. Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We propose a new discretization method for the Stokes equations. The method is an improved version of the method recently presented in [C. Lehrenfeld and J. Sch\u00f6berl, Comp. Meth. Appl. Mech. Eng., 361 (2016)] which is based on an $H({div})$-conforming finite element space and a hybrid discontinuous Galerkin (HDG) formulation of the viscous forces. $H({div})$-conformity results in favorable properties such as pointwise divergence-free solutions and pressure robustness. However, for the approximation of the velocity with a polynomial degree $k$, it requires unknowns of degree $k$ on every facet of the mesh. In view of the superconvergence property of other HDG methods, where only unknowns of polynomial degree $k-1$ on the facets are required to obtain an accurate polynomial approximation of order $k$ (possibly after a local postprocessing), this is suboptimal. The key idea in this paper is to slightly relax the $H({div})$-conformity so that only unknowns of polynomial degree $k-1$ are involved for normal continuity. This allows for optimality of the method also in the sense of superconvergent HDG methods. In order not to lose the benefits of $H({div})$-conformity, we introduce a cheap reconstruction operator which restores pressure robustness and pointwise divergence-free solutions and suits well to the finite element space with relaxed $H({div})$-conformity. We present this new method, carry out a thorough $h$-version error analysis, and demonstrate the performance of the method on numerical examples.<\/jats:p>","DOI":"10.1137\/17m1138078","type":"journal-article","created":{"date-parts":[[2018,7,12]],"date-time":"2018-07-12T20:06:08Z","timestamp":1531425968000},"page":"2070-2094","source":"Crossref","is-referenced-by-count":36,"title":["Hybrid Discontinuous Galerkin Methods with Relaxed H(div)-Conformity for Incompressible Flows. Part I"],"prefix":"10.1137","volume":"56","author":[{"given":"Philip L.","family":"Lederer","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Christoph","family":"Lehrenfeld","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Joachim","family":"Sch\u00f6berl","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,7,12]]},"reference":[{"key":"atypb1","volume-title":"Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables","author":"Abramowitz M.","year":"1974"},{"key":"atypb2","first-page":"293","volume-title":"Encyclopedia of Mathematics and its Applications","author":"Andrews G.","year":"1999"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-012-0461-0"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-006-0681-2"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36519-5"},{"key":"atypb6","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-34797-9","volume-title":"Finite Elemente---Theorie, schnelle L\u00f6ser und Anwendungen in der Elastizit\u00e4tstheorie","author":"Braess D.","year":"2013"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.4208\/jcm.1411-m4499"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1137\/070706616"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-2010-02410-X"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-04-01718-1"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1007\/s10915-006-9107-7"},{"key":"atypb12","first-page":"s018","volume-title":"IMA J. 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