{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,19]],"date-time":"2026-06-19T15:27:27Z","timestamp":1781882847540,"version":"3.54.5"},"reference-count":0,"publisher":"Walter de Gruyter GmbH","issue":"2","funder":[{"DOI":"10.13039\/100006227","name":"Lawrence Livermore National Laboratory","doi-asserted-by":"crossref","award":["DE-AC52-07NA27344"],"award-info":[{"award-number":["DE-AC52-07NA27344"]}],"id":[{"id":"10.13039\/100006227","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2014,4,1]]},"abstract":"<jats:title>Abstract.<\/jats:title>\n               <jats:p>\nWe present two novel coarse spaces (<jats:italic>H<jats:sup>1<\/jats:sup>\n                  <\/jats:italic>- and <jats:inline-formula id=\"eq1_w2aab2b8b9b1b7b1aab1c12b1b3Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0004_81406d89acd036abbfd7e9d2d72199cb.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>H<\/m:mi>\n                              <m:mo>(<\/m:mo>\n                              <m:mo form=\"prefix\">curl<\/m:mo>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$H(\\operatorname{curl})$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-conforming) based on element agglomeration on unstructured tetrahedral meshes. Each <jats:italic>H<jats:sup>1<\/jats:sup>\n                  <\/jats:italic>-conforming coarse basis function is\ncontinuous and piecewise-linear with respect to an original tetrahedral mesh. The <jats:inline-formula id=\"eq2_w2aab2b8b9b1b7b1aab1c12b1b7Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0004_81406d89acd036abbfd7e9d2d72199cb.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>H<\/m:mi>\n                              <m:mo>(<\/m:mo>\n                              <m:mo form=\"prefix\">curl<\/m:mo>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$H(\\operatorname{curl})$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-conforming coarse space is a subspace of the lowest order N\u00e9d\u00e9lec space of the first type. The <jats:italic>H<jats:sup>1<\/jats:sup>\n                  <\/jats:italic>-conforming coarse space exactly interpolates affine functions on each agglomerate. The <jats:inline-formula id=\"eq3_w2aab2b8b9b1b7b1aab1c12b1c11Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0004_81406d89acd036abbfd7e9d2d72199cb.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>H<\/m:mi>\n                              <m:mo>(<\/m:mo>\n                              <m:mo form=\"prefix\">curl<\/m:mo>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$H(\\operatorname{curl})$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-conforming coarse space exactly interpolates vector constants on each agglomerate. Combined with the\n<jats:inline-formula id=\"eq4_w2aab2b8b9b1b7b1aab1c12b1c13Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0004_f505caca7cffabe4355f863b292532fe.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>H<\/m:mi>\n                              <m:mo>(<\/m:mo>\n                              <m:mo form=\"prefix\">div<\/m:mo>\n                              <m:mo>)<\/m:mo>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:tex-math>$H(\\operatorname{div})$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>- and <jats:italic>L<jats:sub>2<\/jats:sub>\n                  <\/jats:italic>-conforming spaces developed previously in\n[Numer. Linear Algebra Appl. 19 (2012), 414\u2013426], the newly\nconstructed coarse spaces form a sequence (with respect to exterior derivatives) which is exact\nas long as the underlying sequence of fine-grid spaces is exact. The constructed coarse spaces inherit the approximation and stability properties of the underlying fine-grid spaces supported by our numerical experiments. The new coarse spaces, in addition to multigrid, can be used for upscaling of broad range of PDEs involving <jats:inline-formula id=\"eq5_w2aab2b8b9b1b7b1aab1c12b1c17Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0004_d079dddd9e7ed94cc068084250bb96b5.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mo form=\"prefix\">curl<\/m:mo>\n                        <\/m:math>\n                        <jats:tex-math>$\\operatorname{curl}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>, <jats:inline-formula id=\"eq6_w2aab2b8b9b1b7b1aab1c12b1c19Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0004_89d08b72372aff629e066938bdbb55c8.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mo form=\"prefix\">div<\/m:mo>\n                        <\/m:math>\n                        <jats:tex-math>$\\operatorname{div}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> and <jats:inline-formula id=\"eq7_w2aab2b8b9b1b7b1aab1c12b1c21Aa\">\n                     <jats:alternatives>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/cmam-2014-0004_d74b9ecfc461a306da46ef8566e84f59.png\"\/>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mo form=\"prefix\">grad<\/m:mo>\n                        <\/m:math>\n                        <jats:tex-math>$\\operatorname{grad}$<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> differential operators.\n<\/jats:p>","DOI":"10.1515\/cmam-2014-0004","type":"journal-article","created":{"date-parts":[[2014,3,13]],"date-time":"2014-03-13T09:23:11Z","timestamp":1394702591000},"page":"257-303","source":"Crossref","is-referenced-by-count":22,"title":["The Construction of the Coarse de Rham Complexes with Improved Approximation Properties"],"prefix":"10.1515","volume":"14","author":[{"given":"Ilya V.","family":"Lashuk","sequence":"first","affiliation":[{"name":"Department of Mathematics, Pennsylvania State University, 212 McAllister Building, University Park, PA 16802, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Panayot S.","family":"Vassilevski","sequence":"additional","affiliation":[{"name":"Center for Applied Scientific Computing, Lawrence Livermore National Laboratory, P.O. Box 808, L-560, Livermore, CA 94550, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"374","published-online":{"date-parts":[[2014,3,13]]},"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/14\/2\/article-p257.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0004\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0004\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T21:32:27Z","timestamp":1680298347000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2014-0004\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2014,3,13]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2014,3,13]]},"published-print":{"date-parts":[[2014,4,1]]}},"alternative-id":["10.1515\/cmam-2014-0004"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2014-0004","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2014,3,13]]}}}