{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T03:38:04Z","timestamp":1740109084671,"version":"3.37.3"},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"2-3","license":[{"start":{"date-parts":[[2022,12,26]],"date-time":"2022-12-26T00:00:00Z","timestamp":1672012800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,12,26]],"date-time":"2022-12-26T00:00:00Z","timestamp":1672012800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100008678","name":"Universit\u00e4t Leipzig","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100008678","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>This paper introduces two methods for the fully discrete time-dependent Bingham problem in a three-dimensional domain and for the flow in a pipe also named after Mosolov. The first time discretisation is a generalised midpoint rule and the second time discretisation is a discontinuous Galerkin scheme. The space discretisation in both cases employs the non-conforming first-order finite elements of Crouzeix and Raviart. The a\u00a0priori error analyses for both schemes yield certain convergence rates in time and optimal convergence rates in space. It guarantees convergence of the fully-discrete scheme with a discontinuous Galerkin time-discretisation for consistent initial conditions and a source term <jats:inline-formula><jats:alternatives><jats:tex-math>$$f\\in H^1(0,T;L^2(\\Omega ))$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                  <mml:mrow>\n                    <mml:mi>f<\/mml:mi>\n                    <mml:mo>\u2208<\/mml:mo>\n                    <mml:msup>\n                      <mml:mi>H<\/mml:mi>\n                      <mml:mn>1<\/mml:mn>\n                    <\/mml:msup>\n                    <mml:mrow>\n                      <mml:mo>(<\/mml:mo>\n                      <mml:mn>0<\/mml:mn>\n                      <mml:mo>,<\/mml:mo>\n                      <mml:mi>T<\/mml:mi>\n                      <mml:mo>\u037e<\/mml:mo>\n                      <mml:msup>\n                        <mml:mi>L<\/mml:mi>\n                        <mml:mn>2<\/mml:mn>\n                      <\/mml:msup>\n                      <mml:mrow>\n                        <mml:mo>(<\/mml:mo>\n                        <mml:mi>\u03a9<\/mml:mi>\n                        <mml:mo>)<\/mml:mo>\n                      <\/mml:mrow>\n                      <mml:mo>)<\/mml:mo>\n                    <\/mml:mrow>\n                  <\/mml:mrow>\n                <\/mml:math><\/jats:alternatives><\/jats:inline-formula>.<\/jats:p>","DOI":"10.1007\/s00211-022-01338-4","type":"journal-article","created":{"date-parts":[[2022,12,26]],"date-time":"2022-12-26T08:02:35Z","timestamp":1672041755000},"page":"411-450","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Two discretisations of the time-dependent Bingham problem"],"prefix":"10.1007","volume":"153","author":[{"given":"C.","family":"Carstensen","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"M.","family":"Schedensack","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,12,26]]},"reference":[{"issue":"4","key":"1338_CR1","doi-asserted-by":"publisher","first-page":"1271","DOI":"10.1137\/S0036142998341301","volume":"37","author":"J Alberty","year":"2000","unstructured":"Alberty, J., Carstensen, C.: Numerical analysis of time-depending primal elastoplasticity with hardening. SIAM J. Numer. Anal. 37(4), 1271\u20131294 (2000)","journal-title":"SIAM J. Numer. Anal."},{"issue":"43","key":"1338_CR2","doi-asserted-by":"publisher","first-page":"4949","DOI":"10.1016\/S0045-7825(02)00422-X","volume":"191","author":"J Alberty","year":"2002","unstructured":"Alberty, J., Carstensen, C.: Discontinuous Galerkin time discretization in elastoplasticity: motivation, numerical algorithms, and applications. Comput. Methods Appl. Mech. Eng. 191(43), 4949\u20134968 (2002)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"215","key":"1338_CR3","doi-asserted-by":"publisher","first-page":"897","DOI":"10.1090\/S0025-5718-96-00746-6","volume":"65","author":"SC Brenner","year":"1996","unstructured":"Brenner, S.C.: Two-level additive Schwarz preconditioners for nonconforming finite element methods. Math. Comp. 65(215), 897\u2013921 (1996)","journal-title":"Math. Comp."},{"issue":"1","key":"1338_CR4","doi-asserted-by":"publisher","first-page":"306","DOI":"10.1137\/S0036142902401311","volume":"41","author":"SC Brenner","year":"2003","unstructured":"Brenner, S.C.: Poincar\u00e9-Friedrichs inequalities for piecewise $$H^{1}$$ functions. SIAM J. Numer. Anal. 41(1), 306\u2013324 (2003)","journal-title":"SIAM J. Numer. Anal."},{"issue":"290","key":"1338_CR5","doi-asserted-by":"publisher","first-page":"2605","DOI":"10.1090\/S0025-5718-2014-02833-0","volume":"83","author":"C Carstensen","year":"2014","unstructured":"Carstensen, C., Gedicke, J.: Guaranteed lower bounds for eigenvalues. Math. Comp. 83(290), 2605\u20132629 (2014)","journal-title":"Math. Comp."},{"issue":"293","key":"1338_CR6","doi-asserted-by":"publisher","first-page":"1061","DOI":"10.1090\/S0025-5718-2014-02894-9","volume":"84","author":"C Carstensen","year":"2015","unstructured":"Carstensen, C., Gallistl, D., Schedensack, M.: Adaptive nonconforming Crouzeix-Raviart FEM for eigenvalue problems. Math. Comp. 84(293), 1061\u20131087 (2015)","journal-title":"Math. Comp."},{"issue":"3","key":"1338_CR7","doi-asserted-by":"publisher","first-page":"1096","DOI":"10.1093\/imanum\/drv051","volume":"36","author":"C Carstensen","year":"2016","unstructured":"Carstensen, C., Gallistl, D., Schedensack, M.: $$L^2$$ best-approximation of the elastic stress in the Arnold-Winther FEM. IMA J. Numer. Anal. 36(3), 1096\u20131119 (2016)","journal-title":"IMA J. Numer. Anal."},{"issue":"3","key":"1338_CR8","doi-asserted-by":"publisher","first-page":"715","DOI":"10.1007\/s00211-018-0999-6","volume":"141","author":"C Carstensen","year":"2019","unstructured":"Carstensen, C., Liu, D., Alberty, J.: Convergence of $$\\rm dG (1)$$ in elastoplastic evolution. Numer. Math. 141(3), 715\u2013742 (2019)","journal-title":"Numer. Math."},{"issue":"6","key":"1338_CR9","doi-asserted-by":"publisher","first-page":"2803","DOI":"10.1137\/110845707","volume":"50","author":"C Carstensen","year":"2012","unstructured":"Carstensen, C., Peterseim, D., Schedensack, M.: Comparison results of finite element methods for the Poisson model problem. SIAM J. Numer. Anal. 50(6), 2803\u20132823 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"1338_CR10","doi-asserted-by":"crossref","unstructured":"Crouzeix, M., Raviart, P.-A.: Conforming and nonconforming finite element methods for solving the stationary Stokes equations. I. Rev. Fran\u00e7aise Automat. Informat. Recherche Op\u00e9rationnelle S\u00e9r. Rouge, 7(R-3):33\u201375, (1973)","DOI":"10.1051\/m2an\/197307R300331"},{"issue":"1","key":"1338_CR11","doi-asserted-by":"publisher","first-page":"37","DOI":"10.1007\/s00211-015-0738-1","volume":"133","author":"C Carstensen","year":"2016","unstructured":"Carstensen, C., Reddy, B.D., Schedensack, M.: A natural nonconforming FEM for the Bingham flow problem is quasi-optimal. Numer. Math. 133(1), 37\u201366 (2016)","journal-title":"Numer. Math."},{"issue":"4","key":"1338_CR12","doi-asserted-by":"publisher","first-page":"1591","DOI":"10.1093\/imanum\/dru048","volume":"35","author":"C Carstensen","year":"2015","unstructured":"Carstensen, C., Schedensack, M.: Medius analysis and comparison results for first-order finite element methods in linear elasticity. IMA J. Numer. Anal. 35(4), 1591\u20131621 (2015)","journal-title":"IMA J. Numer. Anal."},{"key":"1338_CR13","doi-asserted-by":"publisher","first-page":"36","DOI":"10.1016\/j.jnnfm.2006.09.002","volume":"142","author":"EJ Dean","year":"2007","unstructured":"Dean, E.J., Glowinski, R., Guidoboni, G.: On the numerical simulation of Bingham visco-plastic flow: Old and new results. J. Non-Newtonian Fluid Mech. 142, 36\u201362 (2007)","journal-title":"J. Non-Newtonian Fluid Mech."},{"key":"1338_CR14","doi-asserted-by":"crossref","unstructured":"Falk, R.S., Mercier, B.: Error estimates for elastic-plastic problems. RAIRO, Anal. Num\u00e9r., 11:135\u2013144, (1977)","DOI":"10.1051\/m2an\/1977110201351"},{"key":"1338_CR15","doi-asserted-by":"crossref","unstructured":"Gustafsson, T., Lederer, P.L.: Mixed finite elements for Bingham flow in a pipe. Numer. Math. 152(4), 819\u2013840 (2022)","DOI":"10.1007\/s00211-022-01332-w"},{"key":"1338_CR16","doi-asserted-by":"crossref","unstructured":"Glowinski, R.: Sur l\u2019approximation d\u2019une in\u00e9quation variationnelle elliptique de type Bingham. RAIRO Analyse Num\u00e9rique, 10(R-3):13\u201330, 1976","DOI":"10.1051\/m2an\/197610R300131"},{"key":"1338_CR17","doi-asserted-by":"crossref","unstructured":"Glowinski, R.: Numerical methods for nonlinear variational problems. Scientific Computation. Springer-Verlag, Berlin, 2008. Reprint of the 1984 original","DOI":"10.1007\/978-3-662-12613-4"},{"issue":"272","key":"1338_CR18","doi-asserted-by":"publisher","first-page":"2169","DOI":"10.1090\/S0025-5718-10-02360-4","volume":"79","author":"T Gudi","year":"2010","unstructured":"Gudi, T.: A new error analysis for discontinuous finite element methods for linear elliptic problems. Math. Comp. 79(272), 2169\u20132189 (2010)","journal-title":"Math. Comp."},{"key":"1338_CR19","doi-asserted-by":"crossref","unstructured":"Thom\u00e9e, V.: Galerkin finite element methods for parabolic problems. Springer Series in Computational Mathematics, vol. 25. Springer-Verlag, Berlin (1997)","DOI":"10.1007\/978-3-662-03359-3"},{"key":"1338_CR20","unstructured":"Zeidler, E., Nonlinear functional analysis and its applications. III. Springer-Verlag, New York,: Variational methods and optimization. Translated from the German by Leo F, Boron (1985)"},{"issue":"1","key":"1338_CR21","doi-asserted-by":"publisher","first-page":"153","DOI":"10.1007\/s00211-003-0469-6","volume":"96","author":"Y Zhang","year":"2003","unstructured":"Zhang, Y.: Error estimates for the numerical approximation of time-dependent flow of Bingham fluid in cylindrical pipes by the regularization method. Numer. Math. 96(1), 153\u2013184 (2003)","journal-title":"Numer. Math."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01338-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-022-01338-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01338-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,3,26]],"date-time":"2023-03-26T21:18:44Z","timestamp":1679865524000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-022-01338-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,26]]},"references-count":21,"journal-issue":{"issue":"2-3","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["1338"],"URL":"https:\/\/doi.org\/10.1007\/s00211-022-01338-4","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"type":"print","value":"0029-599X"},{"type":"electronic","value":"0945-3245"}],"subject":[],"published":{"date-parts":[[2022,12,26]]},"assertion":[{"value":"6 July 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 November 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 November 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 December 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}