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Anal."],"published-print":{"date-parts":[[2018,1]]},"abstract":"<jats:p>We find new discrete $H^{1}\\!$- and Poincar\u00e9--Friedrichs inequalities by studying the invertibility of the discontinuous Galkerkin (DG) approximation of the flux for local spaces admitting M-decompositions. We then show how to use these inequalities to define and analyze new, superconvergent hybridizable DG (HDG) and mixed methods for which the stabilization function is defined in such a way that the approximations satisfy new $H^1$-stability results with which their error analysis is greatly simplified. We apply this approach to define a wide class of energy-bounded, superconvergent HDG and mixed methods for the incompressible Navier--Stokes equations defined on unstructured meshes using, in two dimensions, general polygonal elements and, in three dimensions, general, flat-faced tetrahedral, prismatic, pyramidal, and hexahedral elements.<\/jats:p>","DOI":"10.1137\/17m1144830","type":"journal-article","created":{"date-parts":[[2018,12,4]],"date-time":"2018-12-04T13:35:19Z","timestamp":1543930519000},"page":"3407-3429","source":"Crossref","is-referenced-by-count":7,"title":["Discrete $H^1$-Inequalities for Spaces Admitting M-Decompositions"],"prefix":"10.1137","volume":"56","author":[{"given":"Bernardo","family":"Cockburn","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Guosheng","family":"Fu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Weifeng","family":"Qiu","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"351","published-online":{"date-parts":[[2018,12,4]]},"reference":[{"key":"atypb1","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36519-5"},{"key":"atypb2","doi-asserted-by":"publisher","DOI":"10.1007\/BF01389710"},{"key":"atypb3","doi-asserted-by":"publisher","DOI":"10.1137\/S0036142900371003"},{"key":"atypb4","doi-asserted-by":"publisher","DOI":"10.1090\/mcom\/3195"},{"key":"atypb5","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drr058"},{"key":"atypb6","doi-asserted-by":"publisher","DOI":"10.1137\/13091573X"},{"key":"atypb7","doi-asserted-by":"publisher","DOI":"10.1137\/080729062"},{"key":"atypb8","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-08-02123-6"},{"key":"atypb9","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/drx025"},{"key":"atypb10","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2016016"},{"key":"atypb11","doi-asserted-by":"publisher","DOI":"10.1051\/m2an\/2016023"},{"key":"atypb12","doi-asserted-by":"publisher","DOI":"10.1137\/16M1073352"},{"key":"atypb13","first-page":"730","volume":"37","author":"Cockburn B.","year":"2017","journal-title":"IMA J. 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