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Both methods use divergence-conforming linear Brezzi\u2013Douglas\u2013Marini space for approximating the velocity and the lowest order Raviart\u2013Thomas space for approximating the vorticity. Our methods are based on the physically correct viscous stress tensor of the fluid, involving the symmetric gradient of velocity (rather than the gradient), provide exactly divergence-free discrete velocity solutions, and optimal error estimates that are also pressure robust. We explain how the methods are constructed using the minimal number of coupling degrees of freedom per facet. The stability analysis of both methods are based on a Korn-like inequality for vector finite elements with continuous normal component. Numerical examples illustrate the theoretical findings and offer comparisons of condition numbers between the two new methods.<\/jats:p>","DOI":"10.1007\/s10915-023-02203-8","type":"journal-article","created":{"date-parts":[[2023,5,11]],"date-time":"2023-05-11T19:02:30Z","timestamp":1683831750000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Divergence-Conforming Velocity and Vorticity Approximations for Incompressible Fluids Obtained with Minimal Facet Coupling"],"prefix":"10.1007","volume":"95","author":[{"given":"J.","family":"Gopalakrishnan","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"L.","family":"Kogler","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-1875-7442","authenticated-orcid":false,"given":"P. 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