{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,13]],"date-time":"2025-11-13T07:14:21Z","timestamp":1763018061971,"version":"3.40.5"},"reference-count":28,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/100000001","name":"National Science Foundation","doi-asserted-by":"publisher","award":["DMS-1217081","DMS-1522707"],"award-info":[{"award-number":["DMS-1217081","DMS-1522707"]}],"id":[{"id":"10.13039\/100000001","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2019,7,1]]},"abstract":"Abstract<\/jats:title>\n This paper develops and analyzes two least-squares methods for the numerical solution\nof linear elasticity and Stokes equations in both two and three dimensions.\nBoth approaches use the \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> norm to define least-squares functionals.\nOne is based on the stress-displacement\/velocity-rotation\/vorticity-pressure (SDRP\/SVVP)\nformulation, and the other is based on the stress-displacement\/velocity-rotation\/vorticity (SDR\/SVV) formulation.\nThe introduction of the rotation\/vorticity variable enables us to weakly enforce the symmetry of the stress.\nIt is shown that the homogeneous least-squares functionals\nare elliptic and continuous in the norm of \n \n \n \n H<\/m:mi>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n div<\/m:mi>\n ;<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H(\\mathrm{div};\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the stress, of \n \n \n \n \n H<\/m:mi>\n 1<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {H^{1}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the\ndisplacement\/velocity, and of \n \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n \u2062<\/m:mo>\n \n (<\/m:mo>\n \u03a9<\/m:mi>\n )<\/m:mo>\n <\/m:mrow>\n <\/m:mrow>\n <\/m:math>\n \n {L^{2}(\\Omega)}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> for the rotation\/vorticity and the pressure.\nThis immediately implies optimal error estimates in the energy norm for conforming finite element\napproximations. As well, it admits optimal multigrid\nsolution methods if Raviart\u2013Thomas finite element spaces are used to approximate the stress tensor.\nThrough a refined duality argument, an optimal \n \n \n \n L<\/m:mi>\n 2<\/m:mn>\n <\/m:msup>\n <\/m:math>\n \n {L^{2}}<\/jats:tex-math>\n <\/jats:alternatives>\n <\/jats:inline-formula> norm error estimates for the displacement\/velocity\nare also established.\nFinally, numerical results for a Cook\u2019s membrane problem of planar elasticity are included in order to illustrate the\nrobustness of our method in the incompressible limit.<\/jats:p>","DOI":"10.1515\/cmam-2018-0255","type":"journal-article","created":{"date-parts":[[2019,6,14]],"date-time":"2019-06-14T09:10:56Z","timestamp":1560503456000},"page":"415-430","source":"Crossref","is-referenced-by-count":6,"title":["Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry"],"prefix":"10.1515","volume":"19","author":[{"given":"Fleurianne","family":"Bertrand","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mathematik , Humboldt-Universit\u00e4t zu Berlin , Unter den Linden 6, 10099 Berlin , Germany"}]},{"given":"Zhiqiang","family":"Cai","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Purdue University , 150 N. University Street , West Lafayette , IN 47907-2067 , USA"}]},{"given":"Eun Young","family":"Park","sequence":"additional","affiliation":[{"name":"Department of Mathematics , Purdue University , 150 N. University Street , West Lafayette , IN 47907-2067 , USA"}]}],"member":"374","published-online":{"date-parts":[[2019,6,13]]},"reference":[{"key":"2025051309565055380_j_cmam-2018-0255_ref_001_w2aab3b7e5292b1b6b1ab2ab1Aa","doi-asserted-by":"crossref","unstructured":"M. Amara and J. M. Thomas,\nEquilibrium finite elements for the linear elastic problem,\nNumer. Math. 33 (1979), no. 4, 367\u2013383.","DOI":"10.1007\/BF01399320"},{"key":"2025051309565055380_j_cmam-2018-0255_ref_002_w2aab3b7e5292b1b6b1ab2ab2Aa","doi-asserted-by":"crossref","unstructured":"D. N. Arnold, G. Awanou and R. Winther,\nFinite elements for symmetric tensors in three dimensions,\nMath. 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