{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,23]],"date-time":"2026-06-23T06:50:38Z","timestamp":1782197438931,"version":"3.54.5"},"reference-count":49,"publisher":"American Mathematical Society (AMS)","issue":"333","license":[{"start":{"date-parts":[[2022,9,28]],"date-time":"2022-09-28T00:00:00Z","timestamp":1664323200000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["647134"],"award-info":[{"award-number":["647134"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>\n                    We study extensions of piecewise polynomial data prescribed in a patch of tetrahedra sharing an edge. We show stability in the sense that the minimizers over piecewise polynomial spaces with prescribed tangential component jumps across faces and prescribed piecewise curl in elements are subordinate in the broken energy norm to the minimizers over the broken\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold-italic upper H left-parenthesis bold c u r l right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi mathvariant=\"bold\">curl<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\boldsymbol H(\\boldsymbol {\\operatorname {curl}})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    space with the same prescriptions. Our proofs are constructive and yield constants independent of the polynomial degree. We then detail the application of this result to the a posteriori error analysis of the curl\u2013curl problem discretized with N\u00e9d\u00e9lec finite elements of arbitrary order. The resulting estimators are reliable, locally efficient, polynomial-degree-robust, and inexpensive. They are constructed by a broken patchwise equilibration which, in particular, does not produce a globally\n                    <inline-formula content-type=\"math\/mathml\">\n                      <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" alttext=\"bold-italic upper H left-parenthesis bold c u r l right-parenthesis\">\n                        <mml:semantics>\n                          <mml:mrow>\n                            <mml:mi mathvariant=\"bold-italic\">H<\/mml:mi>\n                            <mml:mo stretchy=\"false\">(<\/mml:mo>\n                            <mml:mi mathvariant=\"bold\">curl<\/mml:mi>\n                            <mml:mo stretchy=\"false\">)<\/mml:mo>\n                          <\/mml:mrow>\n                          <mml:annotation encoding=\"application\/x-tex\">\\boldsymbol H(\\boldsymbol {\\operatorname {curl}})<\/mml:annotation>\n                        <\/mml:semantics>\n                      <\/mml:math>\n                    <\/inline-formula>\n                    -conforming flux. The equilibration is only related to edge patches and can be realized without solutions of patch problems by a sweep through tetrahedra around every mesh edge. The error estimates become guaranteed when the regularity pick-up constant is explicitly known. Numerical experiments illustrate the theoretical findings.\n                  <\/p>","DOI":"10.1090\/mcom\/3673","type":"journal-article","created":{"date-parts":[[2021,12,10]],"date-time":"2021-12-10T11:41:57Z","timestamp":1639136517000},"page":"37-74","source":"Crossref","is-referenced-by-count":14,"title":["Stable broken \ud835\udc3b(]olf) polynomial extensions and j-robust a posteriori error estimates by broken patchwise equilibration for the curl\u2013curl problem"],"prefix":"10.1090","volume":"91","author":[{"given":"T.","family":"Chaumont-Frelet","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"A.","family":"Ern","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"M.","family":"Vohral\u00edk","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"14","published-online":{"date-parts":[[2021,9,28]]},"reference":[{"key":"1","series-title":"Applied Mathematical Sciences","isbn-type":"print","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-70842-3","volume-title":"Mathematical foundations of computational electromagnetism","volume":"198","author":"Assous, Franck","year":"2018","ISBN":"https:\/\/id.crossref.org\/isbn\/9783319708416"},{"issue":"1","key":"2","doi-asserted-by":"publisher","first-page":"159","DOI":"10.1051\/m2an:2000136","article-title":"Residual based a posteriori error estimators for eddy current computation","volume":"34","author":"Beck, Rudi","year":"2000","journal-title":"M2AN Math. 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