{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,14]],"date-time":"2025-05-14T04:47:21Z","timestamp":1747198041890,"version":"3.40.5"},"reference-count":23,"publisher":"Walter de Gruyter GmbH","issue":"1","license":[{"start":{"date-parts":[[2019,2,15]],"date-time":"2019-02-15T00:00:00Z","timestamp":1550188800000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2020,1,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>The <jats:italic>p<\/jats:italic>-Stokes equations with power-law exponent <jats:inline-formula id=\"j_cmam-2018-0260_ineq_9999\">\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi>p<\/m:mi>\n                              <m:mo>\u2208<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mn>1<\/m:mn>\n                                 <m:mo>,<\/m:mo>\n                                 <m:mn>2<\/m:mn>\n                                 <m:mo stretchy=\"false\">)<\/m:mo>\n                              <\/m:mrow>\n                           <\/m:mrow>\n                        <\/m:math>\n                        <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/j_cmam-2018-0260_eq_0823.png\"\/>\n                        <jats:tex-math>{p\\in(1,2)}<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula> describes non-Newtonian, shear-thinning, incompressible flow. In many industrial applications and natural settings, shear-thinning flow takes place in very thin domains. To account for such anisotropic domains in simulations,\nwe here study an equal-order bi-linear anisotropic finite element discretization of the <jats:italic>p<\/jats:italic>-Stokes equations, and extend a non-linear Local Projection Stabilization to anisotropic meshes. We prove an a priori estimate and illustrate the results with two numerical examples, one confirming the rate of convergence predicted by the a-priori analysis, and one showing the advantages of an anisotropic stabilization compared to an isotropic one.<\/jats:p>","DOI":"10.1515\/cmam-2018-0260","type":"journal-article","created":{"date-parts":[[2019,2,15]],"date-time":"2019-02-15T09:03:43Z","timestamp":1550221423000},"page":"1-25","source":"Crossref","is-referenced-by-count":1,"title":["Equal-Order Stabilized Finite Element Approximation of the <i>p<\/i>-Stokes Equations on Anisotropic Cartesian Meshes"],"prefix":"10.1515","volume":"20","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0329-0253","authenticated-orcid":false,"given":"Josefin","family":"Ahlkrona","sequence":"first","affiliation":[{"name":"Mathematisches Seminar , Christian-Albrechts-Universit\u00e4t zu Kiel , Westring 383, 24118 Kiel , Germany . Current address: Department of Mathematics, Stockholm University, Stockholm, Sweden"}]},{"given":"Malte","family":"Braack","sequence":"additional","affiliation":[{"name":"Mathematisches Seminar , Christian-Albrechts-Universit\u00e4t zu Kiel , Westring 383, 24118 Kiel , Germany"}]}],"member":"374","published-online":{"date-parts":[[2019,2,15]]},"reference":[{"key":"2023033110163477031_j_cmam-2018-0260_ref_001","doi-asserted-by":"crossref","unstructured":"J.  Ahlkrona, N.  Kirchner and P.  L\u00f6tstedt,\nA numerical study of scaling relations for non-Newtonian thin-film flows with applications in ice sheet modelling,\nQuart. J. Mech. Appl. Math. 66 (2013), no. 4, 417\u2013435.","DOI":"10.1093\/qjmam\/hbt009"},{"key":"2023033110163477031_j_cmam-2018-0260_ref_002","unstructured":"T.  Apel,\nAnisotropic Finite Elements: Local Estimates and Applications,\nAdv. Numer. Math.,\nB.\u2009G. Teubner, Stuttgart, 1999."},{"key":"2023033110163477031_j_cmam-2018-0260_ref_003","doi-asserted-by":"crossref","unstructured":"J. W.  Barrett and W. B.  Liu,\nQuasi-norm error bounds for the finite element approximation of a non-Newtonian flow,\nNumer. Math. 68 (1994), no. 4, 437\u2013456.","DOI":"10.1007\/s002110050071"},{"key":"2023033110163477031_j_cmam-2018-0260_ref_004","unstructured":"R.  Becker,\nAn adaptive finite element method for the incompressible Navier\u2013Stokes equations on time-dependent domains,\nPhD thesis, Heidelberg University, 1995."},{"key":"2023033110163477031_j_cmam-2018-0260_ref_005","doi-asserted-by":"crossref","unstructured":"R.  Becker and M.  Braack,\nA finite element pressure gradient stabilization for the Stokes equations based on local projections,\nCalcolo 38 (2001), no. 4, 173\u2013199.","DOI":"10.1007\/s10092-001-8180-4"},{"key":"2023033110163477031_j_cmam-2018-0260_ref_006","doi-asserted-by":"crossref","unstructured":"L.  Belenki, L. C.  Berselli, L.  Diening and M.  R\u016f\u017ei\u010dka,\nOn the finite element approximation of p-Stokes systems,\nSIAM J. Numer. Anal. 50 (2012), no. 2, 373\u2013397.","DOI":"10.1137\/10080436X"},{"key":"2023033110163477031_j_cmam-2018-0260_ref_007","doi-asserted-by":"crossref","unstructured":"J.  Blasco,\nAn anisotropic GLS-stabilized finite element method for incompressible flow problems,\nComput. Methods Appl. Mech. Engrg. 197 (2008), no. 45\u201348, 3712\u20133723.","DOI":"10.1016\/j.cma.2008.02.031"},{"key":"2023033110163477031_j_cmam-2018-0260_ref_008","doi-asserted-by":"crossref","unstructured":"M.  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Num\u00e9r. 27 (1993), no. 2, 131\u2013155.","DOI":"10.1051\/m2an\/1993270201311"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/view\/journals\/cmam\/20\/1\/article-p1.xml","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0260\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0260\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,31]],"date-time":"2023-03-31T11:40:00Z","timestamp":1680262800000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2018-0260\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2019,2,15]]},"references-count":23,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2019,1,20]]},"published-print":{"date-parts":[[2020,1,1]]}},"alternative-id":["10.1515\/cmam-2018-0260"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2018-0260","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2019,2,15]]}}}