{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,15]],"date-time":"2026-04-15T15:03:06Z","timestamp":1776265386568,"version":"3.50.1"},"reference-count":50,"publisher":"Walter de Gruyter GmbH","issue":"3","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2018,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>In this contribution, we review classical mixed methods\nfor the incompressible Navier\u2013Stokes equations that relax the divergence constraint\nand are discretely inf-sup stable. Though the relaxation of the divergence constraint was claimed to be harmless since\nthe beginning of the 1970s,\nPoisson locking is just replaced by another more subtle kind of locking phenomenon, which\nis sometimes called <jats:italic>poor mass conservation<\/jats:italic>\nand led in computational practice to the exclusion\nof mixed methods with low-order pressure approximations like the Bernardi\u2013Raugel\nor the Crouzeix\u2013Raviart finite element methods. Indeed, divergence-free mixed methods\nand classical mixed methods behave\nqualitatively in a different way:\ndivergence-free mixed methods are\n<jats:italic>pressure-robust<\/jats:italic>, which means that, e.g., their velocity error is independent\nof the continuous pressure. The lack of pressure robustness in classical\nmixed methods can be traced back to a consistency error\nof an appropriately defined discrete Helmholtz projector.\nNumerical analysis and numerical examples reveal that\n<jats:italic>really locking-free<\/jats:italic> mixed methods must be discretely inf-sup stable and\npressure-robust, simultaneously. Further, a recent discovery shows that\nlocking-free,\npressure-robust mixed methods do not have to be divergence free.\nIndeed, relaxing the divergence constraint in the velocity trial functions\nis harmless, if the relaxation of the divergence constraint in\nsome velocity test functions is repaired, accordingly.\nThus, inf-sup stable, pressure-robust mixed methods will potentially\nallow in future to reduce the approximation order of the discretizations used in computational\npractice, without compromising the accuracy.<\/jats:p>","DOI":"10.1515\/cmam-2017-0047","type":"journal-article","created":{"date-parts":[[2017,11,18]],"date-time":"2017-11-18T22:15:25Z","timestamp":1511043325000},"page":"353-372","source":"Crossref","is-referenced-by-count":16,"title":["Towards Pressure-Robust Mixed Methods for the Incompressible Navier\u2013Stokes Equations"],"prefix":"10.1515","volume":"18","author":[{"given":"Naveed","family":"Ahmed","sequence":"first","affiliation":[{"name":"Weierstrass Institute , Mohrenstr. 39, 10117 Berlin , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alexander","family":"Linke","sequence":"additional","affiliation":[{"name":"Weierstrass Institute , Mohrenstr. 39, 10117 Berlin , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Christian","family":"Merdon","sequence":"additional","affiliation":[{"name":"Weierstrass Institute , Mohrenstr. 39, 10117 Berlin , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2017,11,18]]},"reference":[{"key":"2023033110284103576_j_cmam-2017-0047_ref_001_w2aab3b7e1250b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"N.  Ahmed, A.  Linke and C.  Merdon,\nTowards pressure-robust mixed methods for the incompressible Navier\u2013Stokes equations,\nFinite Volumes for Complex Applications VIII\u2014Methods and Theoretical Aspects,\nSpringer Proc. Math. Stat. 199,\nSpringer, Cham (2017), 351\u2013359.","DOI":"10.1007\/978-3-319-57397-7_28"},{"key":"2023033110284103576_j_cmam-2017-0047_ref_002_w2aab3b7e1250b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"D. N.  Arnold, F.  Brezzi and M.  Fortin,\nA stable finite element for the Stokes equations,\nCalcolo 21 (1984), no. 4, 337\u2013344.","DOI":"10.1007\/BF02576171"},{"key":"2023033110284103576_j_cmam-2017-0047_ref_003_w2aab3b7e1250b1b6b1ab2ab3Aa","doi-asserted-by":"crossref","unstructured":"I.  Babu\u0161ka and M.  Suri,\nLocking effects in the finite element approximation of elasticity problems,\nNumer. 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