{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,29]],"date-time":"2025-05-29T22:42:45Z","timestamp":1748558565093,"version":"3.37.3"},"reference-count":29,"publisher":"Springer Science and Business Media LLC","issue":"1","license":[{"start":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T00:00:00Z","timestamp":1645056000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T00:00:00Z","timestamp":1645056000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100003725","name":"National Research Foundation of Korea","doi-asserted-by":"publisher","award":["2021R1A2C1003340"],"award-info":[{"award-number":["2021R1A2C1003340"]}],"id":[{"id":"10.13039\/501100003725","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer Algor"],"published-print":{"date-parts":[[2022,9]]},"DOI":"10.1007\/s11075-022-01269-z","type":"journal-article","created":{"date-parts":[[2022,2,17]],"date-time":"2022-02-17T05:02:33Z","timestamp":1645074153000},"page":"449-471","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["A formal construction of a divergence-free basis in the nonconforming virtual element method for the Stokes problem"],"prefix":"10.1007","volume":"91","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5743-1501","authenticated-orcid":false,"given":"Do Y.","family":"Kwak","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hyeokjoo","family":"Park","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,2,17]]},"reference":[{"issue":"3","key":"1269_CR1","doi-asserted-by":"publisher","first-page":"376","DOI":"10.1016\/j.camwa.2013.05.015","volume":"66","author":"B Ahmad","year":"2013","unstructured":"Ahmad, B., Alsaedi, A., Brezzi, F., Marini, L.D., Russo, A.: Equivalent projectors for virtual element methods. Comput. Math. Appl. 66(3), 376\u2013391 (2013)","journal-title":"Comput. Math. Appl."},{"issue":"1","key":"1269_CR2","doi-asserted-by":"publisher","first-page":"386","DOI":"10.1137\/13091141X","volume":"52","author":"PF Antonietti","year":"2014","unstructured":"Antonietti, P.F., Beir\u00e3o da Veiga, L., Mora, D., Verani, M.: A stream virtual element formulation of the Stokes problem on polygonal meshes. SIAM J. Numer. Anal. 52(1), 386\u2013404 (2014)","journal-title":"SIAM J. Numer. Anal."},{"issue":"2","key":"1269_CR3","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1142\/S0218202518500100","volume":"28","author":"PF Antonietti","year":"2018","unstructured":"Antonietti, P.F., Manzini, G., Verani, M.: The fully nonconforming virtual element method for biharmonic problems. Math. Models Methods Appl. Sci. 28(2), 387\u2013407 (2018)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"3","key":"1269_CR4","doi-asserted-by":"publisher","first-page":"879","DOI":"10.1051\/m2an\/2015090","volume":"50","author":"BA de Dios","year":"2016","unstructured":"de Dios, B.A., Lipnikov, K., Manzini, G.: The nonconforming virtual element method. ESAIM Math. Model. Numer. Anal. 50(3), 879\u2013904 (2016)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"1","key":"1269_CR5","doi-asserted-by":"publisher","first-page":"199","DOI":"10.1142\/S0218202512500492","volume":"23","author":"L Beir\u00e3o da Veiga","year":"2013","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Cangiani, A., Manzini, G., Marini, L.D., Russo, A.: Basic principles of virtual element methods. Math. Models Methods Appl. Sci. 23(1), 199\u2013214 (2013)","journal-title":"Math. Models Methods Appl. Sci."},{"key":"1269_CR6","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1016\/j.cma.2017.08.013","volume":"327","author":"L Beir\u00e3o da Veiga","year":"2017","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Dassi, F., Marini, L.D., Russo, A.: Virtual element approximation of 2D magnetostatic problems. Comput. Methods Appl. Mech Engrg. 327, 173\u2013195 (2017)","journal-title":"Comput. Methods Appl. Mech Engrg."},{"issue":"5","key":"1269_CR7","doi-asserted-by":"publisher","first-page":"2940","DOI":"10.1137\/18M1169886","volume":"56","author":"L Beir\u00e3o da Veiga","year":"2018","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Dassi, F., Marini, L.D., Russo, A.: A family of three-dimensional virtual elements with applications to magnetostatics. SIAM J. Numer. Anal. 56(5), 2940\u20132962 (2018)","journal-title":"SIAM J. Numer. Anal."},{"key":"1269_CR8","doi-asserted-by":"publisher","first-page":"343","DOI":"10.1016\/j.cma.2017.12.028","volume":"332","author":"L Beir\u00e3o da Veiga","year":"2018","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Dassi, F., Marini, L.D., Russo, A.: Lowest order virtual element approximation of magnetostatic problems. Comput. Methods Appl. Mech. Engrg. 332, 343\u2013362 (2018)","journal-title":"Comput. Methods Appl. Mech. Engrg."},{"issue":"2","key":"1269_CR9","doi-asserted-by":"publisher","first-page":"794","DOI":"10.1137\/120874746","volume":"51","author":"L Beir\u00e3o da Veiga","year":"2013","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Marini, L.D.: Virtual elements for linear elasticity problems. SIAM J. Numer. Anal. 51(2), 794\u2013812 (2013)","journal-title":"SIAM J. Numer. Anal."},{"issue":"8","key":"1269_CR10","doi-asserted-by":"publisher","first-page":"1541","DOI":"10.1142\/S021820251440003X","volume":"24","author":"L Beir\u00e3o da Veiga","year":"2014","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: The hitchhiker\u2019s guide to the virtual element method. Math. Models Methods Appl. Sci. 24(8), 1541\u20131573 (2014)","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"3","key":"1269_CR11","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1051\/m2an\/2015067","volume":"50","author":"LB da Veiga","year":"2016","unstructured":"da Veiga, L.B., Brezzi, F., Marini, L.D., Russo, A.: Mixed virtual element methods for general second order elliptic problems on polygonal meshes. ESAIM Math. Model. Numer. Anal. 50(3), 727\u2013747 (2016)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"2","key":"1269_CR12","doi-asserted-by":"publisher","first-page":"509","DOI":"10.1051\/m2an\/2016032","volume":"51","author":"LB da Veiga","year":"2017","unstructured":"da Veiga, L.B., Lovadina, C., Vacca, G.: Divergence free virtual elements for the Stokes problem on polygonal meshes. ESAIM Math. Model. Numer. Anal. 51(2), 509\u2013535 (2017)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"key":"1269_CR13","unstructured":"da Veiga, L.B., Dassi, F., Manzini, G., Mascotto, L.: Virtual elements for Maxwell\u2019s equations. Comput. Math. Appl. In press (2021)"},{"issue":"192","key":"1269_CR14","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1090\/S0025-5718-1990-1035927-5","volume":"55","author":"SC Brenner","year":"1990","unstructured":"Brenner, S.C.: A nonconforming multigrid method for the stationary Stokes equations. Math. Comp. 55(192), 411\u2013437 (1990)","journal-title":"Math. Comp."},{"issue":"4","key":"1269_CR15","doi-asserted-by":"publisher","first-page":"1227","DOI":"10.1051\/m2an\/2013138","volume":"48","author":"F Brezzi","year":"2014","unstructured":"Brezzi, F., Falk, R.S.: Basic principles of mixed virtual element methods. ESAIM Math. Model. Numer. Anal. 48(4), 1227\u20131240 (2014)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"6","key":"1269_CR16","doi-asserted-by":"publisher","first-page":"3411","DOI":"10.1137\/15M1049531","volume":"54","author":"A Cangiani","year":"2016","unstructured":"Cangiani, A., Gyrya, V., Manzini, G.: The nonconforming virtual element method for the Stokes equations. SIAM J. Numer. Anal. 54(6), 3411\u20133435 (2016)","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"1269_CR17","first-page":"1317","volume":"37","author":"A Cangiani","year":"2017","unstructured":"Cangiani, A., Manzini, G., Sutton, O.J.: Conforming and nonconforming virtual element methods for elliptic problems. IMA J. Numer. Anal. 37 (3), 1317\u20131354 (2017)","journal-title":"IMA J. Numer. Anal."},{"issue":"R-3","key":"1269_CR18","first-page":"33","volume":"7","author":"M Crouzeix","year":"1973","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)","journal-title":"I. Rev. Fran\u00e7aise Automat. Informat. Recherche Op\u00e9rationnelle S\u00e9r. Rouge"},{"issue":"2","key":"1269_CR19","doi-asserted-by":"publisher","first-page":"303","DOI":"10.1007\/s00211-015-0746-1","volume":"133","author":"L Beir\u00e3o da Veiga","year":"2016","unstructured":"Beir\u00e3o da Veiga, L., Brezzi, F., Marini, L.D., Russo, A.: H(div) and h(curl)-conforming virtual element methods. Numer. Math. 133(2), 303\u2013332 (2016)","journal-title":"Numer. Math."},{"issue":"3","key":"1269_CR20","doi-asserted-by":"publisher","first-page":"749","DOI":"10.1051\/m2an\/2018074","volume":"53","author":"F Gardini","year":"2019","unstructured":"Gardini, F., Manzini, G., Vacca, G.: The nonconforming virtual element method for eigenvalue problems. ESAIM Math. Model. Numer. Anal. 53(3), 749\u2013774 (2019)","journal-title":"ESAIM Math. Model. Numer. Anal."},{"issue":"4","key":"1269_CR21","doi-asserted-by":"publisher","first-page":"2026","DOI":"10.1093\/imanum\/drx063","volume":"38","author":"F Gardini","year":"2018","unstructured":"Gardini, F., Vacca, G.: Virtual element method for second-order elliptic eigenvalue problems. IMA J. Numer. Anal. 38(4), 2026\u20132054 (2018)","journal-title":"IMA J. Numer. Anal."},{"key":"1269_CR22","doi-asserted-by":"crossref","unstructured":"Do, Y., Park, H.: Lowest-order virtual element methods for linear elasticity problems. Comput. Methods Appl. Mech. Kwak Eng. 390:Paper No 114448 (2022)","DOI":"10.1016\/j.cma.2021.114448"},{"key":"1269_CR23","doi-asserted-by":"publisher","first-page":"694","DOI":"10.1016\/j.cma.2017.03.027","volume":"320","author":"X Liu","year":"2017","unstructured":"Liu, X., Li, J., Chen, Z.: A nonconforming virtual element method for the Stokes problem on general meshes. Comput. Methods Appl. Mech. Eng. 320, 694\u2013711 (2017)","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"3","key":"1269_CR24","doi-asserted-by":"publisher","first-page":"309","DOI":"10.1007\/s00158-011-0706-z","volume":"45","author":"C Talischi","year":"2012","unstructured":"Talischi, C., Paulino, G.H., Pereira, A., Menezes, I.F.M.: PolyMesher: a general-purpose mesh generator for polygonal elements written in Matlab. Struct. Multidiscip. Optim. 45(3), 309\u2013328 (2012)","journal-title":"Struct. Multidiscip. Optim."},{"issue":"5","key":"1269_CR25","doi-asserted-by":"publisher","first-page":"1260","DOI":"10.1016\/j.camwa.2020.04.032","volume":"80","author":"X Tang","year":"2020","unstructured":"Tang, X., Liu, Z., Zhang, B., Feng, M.: A low-order locking-free virtual element for linear elasticity problems. Comput. Math. Appl. 80 (5), 1260\u20131274 (2020)","journal-title":"Comput. Math. Appl."},{"key":"1269_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-87047-7","volume-title":"Implementation of finite element methods for Navier-Stokes equations. Springer Series in Computational Physics","author":"F Thomasset","year":"1981","unstructured":"Thomasset, F.: Implementation of finite element methods for Navier-Stokes equations. Springer Series in Computational Physics. Springer, New York-Berlin (1981)"},{"key":"1269_CR27","doi-asserted-by":"publisher","first-page":"394","DOI":"10.1016\/j.jcp.2018.11.004","volume":"378","author":"B Zhang","year":"2019","unstructured":"Zhang, B., Zhao, J., Yang, Y., Chen, S.: The nonconforming virtual element method for elasticity problems. J. Comput. Phys. 378, 394\u2013410 (2019)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"1269_CR28","doi-asserted-by":"publisher","first-page":"610","DOI":"10.1007\/s10915-017-0632-3","volume":"76","author":"J Zhao","year":"2018","unstructured":"Zhao, J., Zhang, B., Chen, S., Mao, S.: The Morley-type virtual element for plate bending problems. J. Sci. Comput. 76(1), 610\u2013629 (2018)","journal-title":"J. Sci. Comput."},{"issue":"6","key":"1269_CR29","doi-asserted-by":"publisher","first-page":"2730","DOI":"10.1137\/18M1200762","volume":"57","author":"J Zhao","year":"2019","unstructured":"Zhao, J., Zhang, B., Mao, S., Chen, S.: The divergence-free nonconforming virtual element for the Stokes problem. SIAM J. Numer. Anal. 57(6), 2730\u20132759 (2019)","journal-title":"SIAM J. Numer. Anal."}],"container-title":["Numerical Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01269-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s11075-022-01269-z\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s11075-022-01269-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,8,8]],"date-time":"2022-08-08T08:20:33Z","timestamp":1659946833000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s11075-022-01269-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,2,17]]},"references-count":29,"journal-issue":{"issue":"1","published-print":{"date-parts":[[2022,9]]}},"alternative-id":["1269"],"URL":"https:\/\/doi.org\/10.1007\/s11075-022-01269-z","relation":{},"ISSN":["1017-1398","1572-9265"],"issn-type":[{"type":"print","value":"1017-1398"},{"type":"electronic","value":"1572-9265"}],"subject":[],"published":{"date-parts":[[2022,2,17]]},"assertion":[{"value":"12 August 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 January 2022","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"17 February 2022","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no competing interests.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"<!--Emphasis Type='Bold' removed-->Conflict of interest"}}]}}