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The DDR complex is fully discrete, meaning that both the spaces and discrete calculus operators are replaced by discrete counterparts, and satisfies suitable exactness properties depending on the topology of the domain. In conjunction with bespoke discrete counterparts of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\\text {L}^2$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:msup>\n                    <mml:mtext>L<\/mml:mtext>\n                    <mml:mn>2<\/mml:mn>\n                  <\/mml:msup>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>-products, it can be used to design schemes for partial differential equations that benefit from the exactness of the sequence but, unlike classical (e.g., Raviart\u2013Thomas\u2013N\u00e9d\u00e9lec) finite elements, are nonconforming. We prove a complete panel of results for the analysis of such schemes: exactness properties, uniform Poincar\u00e9 inequalities, as well as primal and adjoint consistency. We also show how this DDR complex enables the design of a numerical scheme for a magnetostatics problem, and use the aforementioned results to prove stability and optimal error estimates for this scheme.\n<\/jats:p>","DOI":"10.1007\/s10208-021-09542-8","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T17:33:26Z","timestamp":1635874406000},"page":"85-164","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":26,"title":["An Arbitrary-Order Discrete de Rham Complex on Polyhedral Meshes: Exactness, Poincar\u00e9 Inequalities, and Consistency"],"prefix":"10.1007","volume":"23","author":[{"given":"Daniele A.","family":"Di Pietro","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"J\u00e9r\u00f4me","family":"Droniou","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,11,2]]},"reference":[{"issue":"2","key":"9542_CR1","doi-asserted-by":"publisher","first-page":"111","DOI":"10.1515\/cmam-2015-0004","volume":"15","author":"J Aghili","year":"2015","unstructured":"J.\u00a0Aghili, S.\u00a0Boyaval, and D.\u00a0A. 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