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\n On the reference tetrahedron\n \n \n \n \n \n K<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \\widehat K<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , we define three projection-based interpolation operators on\n \n \n \n \n \n H<\/mml:mi>\n 2<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n K<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n H^2(\\widehat K)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n \n \n \n H<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n K<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n \n c<\/mml:mi>\n u<\/mml:mi>\n r<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n {\\mathbf H}^1(\\widehat K,\\operatorname {\\mathbf {curl}})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n \n \n \n \n H<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n 1<\/mml:mn>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n K<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n ,<\/mml:mo>\n div<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n {\\mathbf H}^1(\\widehat K,\\operatorname {div})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . These operators are projections onto spaces of polynomials, they have the commuting diagram property, and they feature the optimal convergence rate as the polynomial degree increases in\n \n \n \n \n \n H<\/mml:mi>\n \n 1<\/mml:mn>\n \n \u2212\n \n <\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n K<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n H^{1-s}(\\widehat K)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n ,\n \n \n \n \n \n \n \n \n H<\/mml:mi>\n <\/mml:mrow>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \n \u2212\n \n <\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n K<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n ,<\/mml:mo>\n \n \n c<\/mml:mi>\n u<\/mml:mi>\n r<\/mml:mi>\n l<\/mml:mi>\n <\/mml:mrow>\n <\/mml:mrow>\n )<\/mml:mo>\n <\/mml:mrow>\n \\widetilde {\\mathbf {H}}^{-s}(\\widehat K,\\operatorname {\\mathbf {curl}})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , and\n \n \n \n \n \n \n \n \n H<\/mml:mi>\n <\/mml:mrow>\n \n ~\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n \n \n \u2212\n \n <\/mml:mo>\n s<\/mml:mi>\n <\/mml:mrow>\n <\/mml:msup>\n (<\/mml:mo>\n \n \n K<\/mml:mi>\n \n ^\n \n <\/mml:mo>\n <\/mml:mover>\n <\/mml:mrow>\n ,<\/mml:mo>\n div<\/mml:mi>\n )<\/mml:mo>\n <\/mml:mrow>\n \\widetilde {\\mathbf {H}}^{-s}(\\widehat K,\\operatorname {div})<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n for\n \n \n \n \n 0<\/mml:mn>\n \n \u2264\n \n <\/mml:mo>\n s<\/mml:mi>\n \n \u2264\n \n <\/mml:mo>\n 1<\/mml:mn>\n <\/mml:mrow>\n 0 \\leq s \\leq 1<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n .\n <\/p>","DOI":"10.1090\/mcom\/3454","type":"journal-article","created":{"date-parts":[[2019,4,25]],"date-time":"2019-04-25T12:33:01Z","timestamp":1556195581000},"page":"45-87","source":"Crossref","is-referenced-by-count":18,"title":["On commuting \ud835\udc5d-version projection-based interpolation on tetrahedra"],"prefix":"10.1090","volume":"89","author":[{"given":"J.","family":"Melenk","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"C.","family":"Rojik","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2019,6,11]]},"reference":[{"issue":"5","key":"1","doi-asserted-by":"publisher","first-page":"640","DOI":"10.1002\/mana.200610762","article-title":"Explicit polynomial preserving trace liftings on a triangle","volume":"282","author":"Ainsworth, Mark","year":"2009","journal-title":"Math. 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