{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,15]],"date-time":"2026-01-15T12:44:02Z","timestamp":1768481042694,"version":"3.49.0"},"reference-count":37,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T00:00:00Z","timestamp":1663286400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T00:00:00Z","timestamp":1663286400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100011914","name":"M\u00e4lardalen University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100011914","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2022,10]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>Degenerating tetrahedral partitions show up quite often in modern finite element analysis. Actually the commonly used maximum angle condition allows some types of element degeneracies. Also, mesh generators and various adaptive procedures may easily produce degenerating mesh elements. Finally, complicated forms of computational domains (e.g. along with a priori known solution layers, etc) may demand the usage of elements of various degenerating shapes. In this paper, we show that the maximum angle condition presents a threshold property in interpolation theory, as the interpolation error may grow (or at least does not decay) if this condition is violated (which does not necessarily imply that FEM error grows). We also demonstrate that the popular red refinements, if done inappropriately, may lead to degenerating partitions which break the maximum angle condition. Finally, we prove that not all tetrahedral elements from a family of tetrahedral partitions are badly shaped when the discretization parameter tends to zero.<\/jats:p>","DOI":"10.1007\/s00211-022-01317-9","type":"journal-article","created":{"date-parts":[[2022,9,16]],"date-time":"2022-09-16T07:34:04Z","timestamp":1663313644000},"page":"307-329","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":2,"title":["On degenerating finite element tetrahedral partitions"],"prefix":"10.1007","volume":"152","author":[{"given":"Sergey","family":"Korotov","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Michal","family":"K\u0159\u00ed\u017eek","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"V\u00e1clav","family":"Ku\u010dera","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,9,16]]},"reference":[{"key":"1317_CR1","doi-asserted-by":"publisher","first-page":"214","DOI":"10.1137\/0713021","volume":"13","author":"I Babu\u0161ka","year":"1976","unstructured":"Babu\u0161ka, I., Aziz, A.K.: On the angle condition in the finite element method. SIAM J. Numer. Anal. 13, 214\u2013226 (1976)","journal-title":"SIAM J. Numer. Anal."},{"key":"1317_CR2","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s002110050475","volume":"85","author":"J Bey","year":"2000","unstructured":"Bey, J.: Simplicial grid refinement: on Freudenthal\u2019s algorithm and the optimal number of congruence classes. Numer. Math. 85, 1\u201329 (2000)","journal-title":"Numer. Math."},{"key":"1317_CR3","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-55677-8","volume-title":"Simplicial Partitions with Applications to the Finite Element Method","author":"J Brandts","year":"2020","unstructured":"Brandts, J., Korotov, S., K\u0159\u00ed\u017eek, M.: Simplicial Partitions with Applications to the Finite Element Method. Springer, Berlin (2020)"},{"key":"1317_CR4","doi-asserted-by":"crossref","unstructured":"Cheng, S.W., Dey, T.K., Edelsbrunner, H., Facello, M.A., Teng, S.H., Sliver exudation. In: Proceedings of 15-th ACM Symposium Computational Geometry, pp. 1\u201313 (1999)","DOI":"10.1145\/304893.304894"},{"key":"1317_CR5","volume-title":"The Finite Element Method for Elliptic Problems","author":"PG Ciarlet","year":"1978","unstructured":"Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. North-Holland, Amsterdam (1978)"},{"key":"1317_CR6","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1137\/0910064","volume":"10","author":"EF D\u2019Azevedo","year":"1989","unstructured":"D\u2019Azevedo, E.F., Simpson, R.B.: On optimal interpolation triangle incidences. SIAM J. Sci. Stat. Comput. 10, 1063\u20131075 (1989)","journal-title":"SIAM J. Sci. Stat. Comput."},{"key":"1317_CR7","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/BF01385784","volume":"59","author":"EF D\u2019Azevedo","year":"1991","unstructured":"D\u2019Azevedo, E.F., Simpson, R.B.: On optimal triangular meshes for minimizing the gradient error. Numer. Math. 59, 321\u2013348 (1991)","journal-title":"Numer. Math."},{"key":"1317_CR8","doi-asserted-by":"publisher","first-page":"133","DOI":"10.1017\/S0962492900001331","volume":"9","author":"H Edelsbrunner","year":"2000","unstructured":"Edelsbrunner, H.: Triangulations and meshes in computational geometry. Acta Numer. 9, 133\u2013213 (2000)","journal-title":"Acta Numer."},{"key":"1317_CR9","doi-asserted-by":"publisher","first-page":"184","DOI":"10.1080\/0025570X.1990.11977515","volume":"63","author":"F Eriksson","year":"1990","unstructured":"Eriksson, F.: On the measure of solid angles. Math. Mag. 63, 184\u2013187 (1990)","journal-title":"Math. Mag."},{"key":"1317_CR10","doi-asserted-by":"publisher","first-page":"370","DOI":"10.1080\/00029890.1952.11988143","volume":"59","author":"JW Gaddum","year":"1952","unstructured":"Gaddum, J.W.: The sum of dihedral and trihedral angles in a tetrahedron. Am. Math. Mon. 59, 370\u2013371 (1952)","journal-title":"Am. Math. Mon."},{"key":"1317_CR11","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1090\/mcom\/3383","volume":"88","author":"J Grande","year":"2019","unstructured":"Grande, J.: Red-green refinement of simplicial meshes in $$d$$ dimensions. Math. Comput. 88, 751\u2013782 (2019)","journal-title":"Math. Comput."},{"key":"1317_CR12","doi-asserted-by":"publisher","first-page":"79","DOI":"10.1007\/s00211-011-0403-2","volume":"120","author":"A Hannukainen","year":"2012","unstructured":"Hannukainen, A., Korotov, S., K\u0159\u00ed\u017eek, M.: The maximum angle condition is not necessary for convergence of the finite element method. Numer. Math. 120, 79\u201388 (2012)","journal-title":"Numer. Math."},{"key":"1317_CR13","first-page":"276","volume":"1","author":"W Huang","year":"2006","unstructured":"Huang, W.: Mathematical principles of anisotropic mesh adaptation. Commun. Comput. Phys. 1, 276\u2013310 (2006)","journal-title":"Commun. Comput. Phys."},{"key":"1317_CR14","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/j.cam.2019.03.003","volume":"358","author":"A Khademi","year":"2019","unstructured":"Khademi, A., Korotov, S., Vatne, J.E.: On the generalization of the Synge\u2013K\u0159\u00ed\u017eek maximum angle condition for $$d$$-simplices. J. Comput. Appl. Math. 358, 29\u201333 (2019)","journal-title":"J. Comput. Appl. Math."},{"key":"1317_CR15","doi-asserted-by":"publisher","first-page":"485","DOI":"10.1007\/s10492-015-0108-4","volume":"60","author":"K Kobayashi","year":"2015","unstructured":"Kobayashi, K., Tsuchiya, T.: A priori error estimates for Lagrange interpolation on triangles. Appl. Math. 60, 485\u2013499 (2015)","journal-title":"Appl. Math."},{"key":"1317_CR16","doi-asserted-by":"publisher","first-page":"105302","DOI":"10.1016\/j.jat.2019.105302","volume":"249","author":"K Kobayashi","year":"2020","unstructured":"Kobayashi, K., Tsuchiya, T.: Error analysis of Lagrange interpolation on tetrahedrons. J. Approx. Theory 249, 105302\u201320 (2020)","journal-title":"J. Approx. Theory"},{"key":"1317_CR17","doi-asserted-by":"publisher","first-page":"2199","DOI":"10.1016\/j.camwa.2014.01.025","volume":"67","author":"S Korotov","year":"2014","unstructured":"Korotov, S., K\u0159\u00ed\u017eek, M.: Red refinements of simplices into congruent subsimplices. Comput. Math. Appl. 67, 2199\u20132204 (2014)","journal-title":"Comput. Math. Appl."},{"key":"1317_CR18","doi-asserted-by":"crossref","unstructured":"Korotov, S., K\u0159\u00ed\u017eek, M., On degenerating tetrahedra resulting from red refinements of tetrahedral partitions. Numer. Anal. Appl. 1\u201310 (2021)","DOI":"10.1134\/S1995423921040030"},{"key":"1317_CR19","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/j.cam.2015.03.046","volume":"293","author":"S Korotov","year":"2016","unstructured":"Korotov, S., Plaza, \u00c1., Su\u00e1rez, J.: Longest-edge $$n$$-section algorithms: properties and open problems. J. Comput. Appl. Math. 293, 139\u2013146 (2016)","journal-title":"J. Comput. Appl. Math."},{"key":"1317_CR20","doi-asserted-by":"crossref","first-page":"46","DOI":"10.21136\/AM.1982.103944","volume":"27","author":"M K\u0159\u00ed\u017eek","year":"1982","unstructured":"K\u0159\u00ed\u017eek, M.: An equilibrium finite element method in three-dimensional elasticity. Apl. Mat. 27, 46\u201375 (1982)","journal-title":"Apl. Mat."},{"key":"1317_CR21","doi-asserted-by":"publisher","first-page":"513","DOI":"10.1137\/0729031","volume":"29","author":"M K\u0159\u00ed\u017eek","year":"1992","unstructured":"K\u0159\u00ed\u017eek, M.: On the maximum angle condition for linear tetrahedral elements. SIAM J. Numer. Anal. 29, 513\u2013520 (1992)","journal-title":"SIAM J. Numer. Anal."},{"key":"1317_CR22","doi-asserted-by":"publisher","first-page":"201","DOI":"10.1002\/(SICI)1098-2426(199703)13:2<201::AID-NUM5>3.0.CO;2-T","volume":"13","author":"M K\u0159\u00ed\u017eek","year":"1997","unstructured":"K\u0159\u00ed\u017eek, M., Strouboulis, T.: How to generate local refinements of unstructured tetrahedral meshes satisfying a regularity ball condition. Numer. Methods Part. Differ. Equ. 13, 201\u2013214 (1997)","journal-title":"Numer. Methods Part. Differ. Equ."},{"key":"1317_CR23","first-page":"132","volume":"2015","author":"V Ku\u010dera","year":"2015","unstructured":"Ku\u010dera, V.: A note on necessary and sufficient condition for convergence of the finite element method. Appl. Math. 2015, 132\u2013139 (2015)","journal-title":"Appl. Math."},{"key":"1317_CR24","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/s10492-016-0132-z","volume":"61","author":"V Ku\u010dera","year":"2016","unstructured":"Ku\u010dera, V.: Several notes on the circumradius condition. Appl. Math. 61, 287\u2013298 (2016)","journal-title":"Appl. Math."},{"key":"1317_CR25","doi-asserted-by":"publisher","first-page":"268","DOI":"10.1007\/BF01955874","volume":"34","author":"A Liu","year":"1994","unstructured":"Liu, A., Joe, B.: Relatioship between tetrahedron shape measures. BIT 34, 268\u2013287 (1994)","journal-title":"BIT"},{"key":"1317_CR26","doi-asserted-by":"publisher","first-page":"1183","DOI":"10.1090\/S0025-5718-96-00748-X","volume":"65","author":"A Liu","year":"1996","unstructured":"Liu, A., Joe, B.: Quality local refinement of tetrahedral meshes based on 8-subtetrahedron subdivision. Math. Comp. 65, 1183\u20131200 (1996)","journal-title":"Math. Comp."},{"key":"1317_CR27","doi-asserted-by":"publisher","first-page":"329","DOI":"10.1016\/j.cam.2008.11.008","volume":"230","author":"S Mao","year":"2009","unstructured":"Mao, S., Shi, Z.: Error estimates of triangular finite elements under a weak angle condition. J. Comput. Appl. Math. 230, 329\u2013331 (2009)","journal-title":"J. Comput. Appl. Math."},{"key":"1317_CR28","doi-asserted-by":"publisher","first-page":"1134","DOI":"10.1137\/0915070","volume":"15","author":"MEG Ong","year":"1994","unstructured":"Ong, M.E.G.: Uniform refinement of a tetrahedron. SIAM J. Sci. Comput. 15, 1134\u20131144 (1994)","journal-title":"SIAM J. Sci. Comput."},{"key":"1317_CR29","doi-asserted-by":"publisher","first-page":"473","DOI":"10.1007\/s10492-015-0107-5","volume":"60","author":"P Oswald","year":"2015","unstructured":"Oswald, P.: Divergence of FEM: Babu\u0161ka\u2013Aziz triangulations revisited. Appl. Math. 60, 473\u2013484 (2015)","journal-title":"Appl. Math."},{"key":"1317_CR30","doi-asserted-by":"publisher","first-page":"2538","DOI":"10.1137\/10081842X","volume":"50","author":"A Rand","year":"2012","unstructured":"Rand, A.: Average interpolation under the maximum angle condition. SIAM J. Numer. Anal. 50, 2538\u20132559 (2012)","journal-title":"SIAM J. Numer. Anal."},{"key":"1317_CR31","doi-asserted-by":"publisher","DOI":"10.1007\/978-94-015-8308-4","volume-title":"Survey of Applicable Mathematics","author":"K Rektorys","year":"1994","unstructured":"Rektorys, K.: Survey of Applicable Mathematics, vol. 1. Kluwer Academic Publishers, Dordrecht (1994)"},{"key":"1317_CR32","doi-asserted-by":"publisher","first-page":"257","DOI":"10.1137\/0729017","volume":"29","author":"S Rippa","year":"1992","unstructured":"Rippa, S.: Long and thin triangles can be good for linear interpolation. SIAM J. Numer. Anal. 29, 257\u2013270 (1992)","journal-title":"SIAM J. Numer. Anal."},{"key":"1317_CR33","doi-asserted-by":"publisher","first-page":"81","DOI":"10.1007\/BF01395933","volume":"19","author":"G Strang","year":"1971","unstructured":"Strang, G.: Approximation in the finite element method. Numer. Math. 19, 81\u201398 (1971)","journal-title":"Numer. Math."},{"key":"1317_CR34","volume-title":"An Analysis of the Finite Element Method","author":"G Strang","year":"1973","unstructured":"Strang, G., Fix, G.: An Analysis of the Finite Element Method. Prentice-Hall Inc, Englewood Cliffs (1973)"},{"key":"1317_CR35","doi-asserted-by":"publisher","DOI":"10.1063\/1.3060143","volume-title":"The Hypercircle in Mathematical Physics","author":"JL Synge","year":"1957","unstructured":"Synge, J.L.: The Hypercircle in Mathematical Physics. Cambridge Univ. Press, Cambridge (1957)"},{"key":"1317_CR36","doi-asserted-by":"publisher","first-page":"355","DOI":"10.21136\/AM.1969.103246","volume":"14","author":"A \u017den\u00ed\u0161ek","year":"1969","unstructured":"\u017den\u00ed\u0161ek, A.: Convergence of the finite element method for boundary value problems of a system of elliptic equations (in Czech). Appl. Math. 14, 355\u2013377 (1969)","journal-title":"Appl. Math."},{"key":"1317_CR37","first-page":"541","volume":"21","author":"S Zhang","year":"1995","unstructured":"Zhang, S.: Successive subdivisions of tetrahedra and multigrid methods on tetrahedral meshes. Houston J. Math. 21, 541\u2013556 (1995)","journal-title":"Houston J. Math."}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01317-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-022-01317-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-022-01317-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,10,3]],"date-time":"2024-10-03T23:59:18Z","timestamp":1727999958000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-022-01317-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,9,16]]},"references-count":37,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,10]]}},"alternative-id":["1317"],"URL":"https:\/\/doi.org\/10.1007\/s00211-022-01317-9","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,9,16]]},"assertion":[{"value":"28 May 2021","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"23 April 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 September 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 September 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}