{"id":751,"date":"2019-10-02T10:29:16","date_gmt":"2019-10-02T08:29:16","guid":{"rendered":"http:\/\/project.inria.fr\/gatipor\/?p=751"},"modified":"2021-12-02T22:06:35","modified_gmt":"2021-12-02T21:06:35","slug":"local-and-global-best-equivalence-in-hdiv-simple-projector-and-optimal-hp-approximation","status":"publish","type":"post","link":"https:\/\/project.inria.fr\/gatipor\/local-and-global-best-equivalence-in-hdiv-simple-projector-and-optimal-hp-approximation\/","title":{"rendered":"Local- and global-best equivalence, simple projector, and optimal <em>hp<\/em> approximation in <strong><em>H<\/em><\/strong>(div)"},"content":{"rendered":"<table style=\"width: 100%;\">\n<tbody>\n<tr>\n<td>\n<p><div id=\"attachment_140\" style=\"width: 910px\" class=\"wp-caption alignnone\"><a href=\"http:\/\/project.inria.fr\/gatipor\/files\/2019\/10\/loc_glob_div.png\"><img decoding=\"async\" aria-describedby=\"caption-attachment-140\" src=\"http:\/\/project.inria.fr\/gatipor\/files\/2019\/10\/loc_glob_div.png\" alt=\"local_global\" width=\"900\" class=\"alignnone size-large wp-image-452\" \/><\/a><p id=\"caption-attachment-140\" class=\"wp-caption-text\">For any <strong><em>H<\/em><\/strong>(div) function <strong><em>v<\/em><\/strong> with zero normal flux over part of the boundary and piecewise polynomial divergence<\/p><\/div><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p><strong>Main results<\/strong>:<\/p>\n<ul>\n<li> <strong>global-best approximation<\/strong> (constraints on normal component continuity and divergence) is <strong>equivalent<\/strong> to the sum of independent <strong>local-best approximations<\/strong> (no constraints);<\/li>\n<li>gives rise to a <strong>simple stable local commuting projector<\/strong> in <strong><em>H<\/em><\/strong>(div);<\/li>\n<li>this projector delivers approximation error equivalent to the local-best approximation;<\/li>\n<li>leads to <strong>optimal<\/strong> <em>hp<\/em> <strong>approximation estimates<\/strong> in <strong><em>H<\/em><\/strong>(div);<\/li>\n<li>applies under the <strong>minimal<\/strong> necessary Sobolev <strong>regularity<\/strong>;<\/li>\n<li>gives optimal a priori <em>hp<\/em>-error estimates for mixed and least-squares finite element methods.<\/li>\n<\/ul>\n<p>Details: <\/p>\n<ul>\n<li><a href=\"https:\/\/academic.oup.com\/imajna\/advance-article\/doi\/10.1093\/imanum\/draa103\/6171135\">paper<\/a> (<a href=\"https:\/\/hal.inria.fr\/hal-02268960\">preprint<\/a>) with Alexandre Ern, Thirupathi Gudi, and Iain Smears, see also <a href=\"https:\/\/who.rocq.inria.fr\/Martin.Vohralik\/Files\/Exp_USNCCM_19.pdf\">presentation<\/a>;<\/li>\n<li>local- and global-best equivalence in <strong><em>H<\/em><\/strong>(curl): <a href=\"https:\/\/rdcu.be\/cBF5g\">paper<\/a> (<a href=\"https:\/\/hal.inria.fr\/hal-02736200\">preprint<\/a>) with Th\u00e9ophile Chaumont-Frelet.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>Main results: global-best approximation (constraints on normal component continuity and divergence) is equivalent to the sum of independent local-best approximations (no constraints); gives rise to a simple stable local commuting projector in H(div); this projector delivers approximation error equivalent to the local-best approximation; leads to optimal hp approximation estimates in\u2026<\/p>\n<p> <a class=\"continue-reading-link\" href=\"https:\/\/project.inria.fr\/gatipor\/local-and-global-best-equivalence-in-hdiv-simple-projector-and-optimal-hp-approximation\/\"><span>Continue reading<\/span><i class=\"crycon-right-dir\"><\/i><\/a> <\/p>\n","protected":false},"author":932,"featured_media":758,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-751","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-uncategorized"],"_links":{"self":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/751","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/users\/932"}],"replies":[{"embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/comments?post=751"}],"version-history":[{"count":14,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/751\/revisions"}],"predecessor-version":[{"id":1054,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/posts\/751\/revisions\/1054"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/media\/758"}],"wp:attachment":[{"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/media?parent=751"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/categories?post=751"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/project.inria.fr\/gatipor\/wp-json\/wp\/v2\/tags?post=751"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}