{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2023,10,5]],"date-time":"2023-10-05T10:53:37Z","timestamp":1696503217749},"reference-count":32,"publisher":"Walter de Gruyter GmbH","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2023,10,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>Recently, <jats:inline-formula>\n                     <jats:alternatives>\n                        <m:math xmlns:m=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                           <m:mrow>\n                              <m:mi mathvariant=\"bold\">H<\/m:mi>\n                              <m:mo>\u2062<\/m:mo>\n                              <m:mrow>\n                                 <m:mo stretchy=\"false\">(<\/m:mo>\n                                 <m:mi>div<\/m:mi>\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-2022-0235_ineq_0001.png\" \/>\n                        <jats:tex-math>\\mathbf{H}(\\mathrm{div})<\/jats:tex-math>\n                     <\/jats:alternatives>\n                  <\/jats:inline-formula>-conforming finite element families were proven to be successful on anisotropic meshes, with the help of suitable interpolation error estimates.\nIn order to ensure corresponding large-scale computation, this contribution provides novel Raviart\u2013Thomas basis functions, robust regarding the anisotropy of a given triangulation.\nThis new set of basis functions on simplices uses a hierarchical approach, and the orientation of the basis functions is inherited from the lowest-order case.\nIn the higher-order case, the new basis functions can be written as a combination of the lowest-order Raviart\u2013Thomas elements and higher-order Lagrange-elements.\nThis ensures robustness regarding the mesh anisotropy and assembling strategies as demonstrated in the numerical experiments.<\/jats:p>","DOI":"10.1515\/cmam-2022-0235","type":"journal-article","created":{"date-parts":[[2023,9,18]],"date-time":"2023-09-18T09:41:20Z","timestamp":1695030080000},"page":"831-847","source":"Crossref","is-referenced-by-count":1,"title":["Novel Raviart\u2013Thomas Basis Functions on Anisotropic Finite Elements"],"prefix":"10.1515","volume":"23","author":[{"given":"Fleurianne","family":"Bertrand","sequence":"first","affiliation":[{"name":"TU Chemnitz , Chemnitz , Germany"}]}],"member":"374","published-online":{"date-parts":[[2023,9,19]]},"reference":[{"key":"2023100411150468233_j_cmam-2022-0235_ref_001","doi-asserted-by":"crossref","unstructured":"G. Acosta, T. Apel, R. G. Dur\u00e1n and A. L. Lombardi,\nError estimates for Raviart\u2013Thomas interpolation of any order on anisotropic tetrahedra,\nMath. Comp. 80 (2011), no. 273, 141\u2013163.","DOI":"10.1090\/S0025-5718-2010-02406-8"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_002","doi-asserted-by":"crossref","unstructured":"G. Acosta and R. G. Dur\u00e1n,\nThe maximum angle condition for mixed and nonconforming elements: Application to the Stokes equations,\nSIAM J. Numer. Anal. 37 (1999), no. 1, 18\u201336.","DOI":"10.1137\/S0036142997331293"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_003","unstructured":"T. Apel,\nAnisotropic Finite Elements: Local Estimates and Applications,\nAdv. Numer. Math.,\nB.\u2009G. Teubner, Stuttgart, 1999."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_004","doi-asserted-by":"crossref","unstructured":"T. Apel and M. Dobrowolski,\nAnisotropic interpolation with applications to the finite element method,\nComputing 47 (1992), no. 3\u20134, 277\u2013293.","DOI":"10.1007\/BF02320197"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_005","doi-asserted-by":"crossref","unstructured":"T. Apel and V. Kempf,\nBrezzi\u2013Douglas\u2013Marini interpolation of any order on anisotropic triangles and tetrahedra,\nSIAM J. Numer. Anal. 58 (2020), no. 3, 1696\u20131718.","DOI":"10.1137\/19M1302910"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_006","doi-asserted-by":"crossref","unstructured":"D. N. Arnold, R. S. Falk and R. Winther,\nFinite element exterior calculus, homological techniques, and applications,\nActa Numer. 15 (2006), 1\u2013155.","DOI":"10.1017\/S0962492906210018"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_007","doi-asserted-by":"crossref","unstructured":"I. Babu\u0161ka and A. K. Aziz,\nOn the angle condition in the finite element method,\nSIAM J. Numer. Anal. 13 (1976), no. 2, 214\u2013226.","DOI":"10.1137\/0713021"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_008","doi-asserted-by":"crossref","unstructured":"C. Bahriawati and C. Carstensen,\nThree MATLAB implementations of the lowest-order Raviart\u2013Thomas MFEM with a posteriori error control,\nComput. Methods Appl. Math. 5 (2005), no. 4, 333\u2013361.","DOI":"10.2478\/cmam-2005-0016"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_009","doi-asserted-by":"crossref","unstructured":"F. Bertrand,\nFirst-order system least-squares for interface problems,\nSIAM J. Numer. Anal. 56 (2018), no. 3, 1711\u20131730.","DOI":"10.1137\/16M1105827"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_010","doi-asserted-by":"crossref","unstructured":"F. Bertrand, B. Kober, M. Moldenhauer and G. Starke,\nEquilibrated stress reconstruction and a posteriori error estimation for linear elasticity,\nInternational Centre for Mechanical Sciences, Courses and Lectures,\nCISM Courses and Lect. 507,\nSpringer, Cham (2020), 69\u2013106.","DOI":"10.1007\/978-3-030-33520-5_3"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_011","doi-asserted-by":"crossref","unstructured":"F. Bertrand, M. Moldenhauer and G. Starke,\nA posteriori error estimation for planar linear elasticity by stress reconstruction,\nComput. Methods Appl. Math. 19 (2019), no. 3, 663\u2013679.","DOI":"10.1515\/cmam-2018-0004"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_012","doi-asserted-by":"crossref","unstructured":"F. Bertrand, M. Moldenhauer and G. Starke,\nWeakly symmetric stress equilibration for hyperelastic material models,\nGAMM-Mitt. 43 (2020), no. 2, Article ID e202000007.","DOI":"10.1002\/gamm.202000007"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_013","doi-asserted-by":"crossref","unstructured":"F. Bertrand, S. M\u00fcnzenmaier and G. Starke,\nFirst-order system least squares on curved boundaries: Higher-order Raviart\u2013Thomas elements,\nSIAM J. Numer. Anal. 52 (2014), no. 6, 3165\u20133180.","DOI":"10.1137\/130948902"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_014","doi-asserted-by":"crossref","unstructured":"F. Bertrand, S. M\u00fcnzenmaier and G. Starke,\nFirst-order system least squares on curved boundaries: Lowest-order Raviart\u2013Thomas elements,\nSIAM J. Numer. Anal. 52 (2014), no. 2, 880\u2013894.","DOI":"10.1137\/13091720X"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_015","doi-asserted-by":"crossref","unstructured":"F. Bertrand and G. Starke,\nA posteriori error estimates by weakly symmetric stress reconstruction for the Biot problem,\nComput. Math. Appl. 91 (2021), 3\u201316.","DOI":"10.1016\/j.camwa.2020.10.011"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_016","doi-asserted-by":"crossref","unstructured":"S. Beuchler, V. Pillwein and S. Zaglmayr,\nSparsity optimized high order finite element functions for \n                  \n                     \n                        \n                           H\n                           \u2062\n                           \n                              (\n                              \n                                 c\n                                 \u2062\n                                 u\n                                 \u2062\n                                 r\n                                 \u2062\n                                 l\n                              \n                              )\n                           \n                        \n                     \n                     \n                     H(curl)\n                  \n                on tetrahedra,\nAdv. Appl. Math. 50 (2013), no. 5, 749\u2013769.","DOI":"10.1016\/j.aam.2012.11.004"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_017","doi-asserted-by":"crossref","unstructured":"D. Boffi, F. Brezzi and M. Fortin,\nMixed Finite Element Methods and Applications,\nSpringer Ser. Comput. Math. 44,\nSpringer, Heidelberg, 2013.","DOI":"10.1007\/978-3-642-36519-5"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_018","doi-asserted-by":"crossref","unstructured":"C. Carstensen, M. Feischl, M. Page and D. Praetorius,\nAxioms of adaptivity,\nComput. Math. Appl. 67 (2014), no. 6, 1195\u20131253.","DOI":"10.1016\/j.camwa.2013.12.003"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_019","unstructured":"F. Cavallero, A. Pieraccini and S. Scial\u00f2,\nA stabilized Raviart\u2013Thomas method for the Navier\u2013Stokes equations with non-linear viscosity,\nComput. & Fluids 219 (2021), Article ID 104986."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_020","unstructured":"V. J. Ervin,\nComputational bases for \n                  \n                     \n                        \n                           RT\n                           k\n                        \n                     \n                     \n                     \\mathrm{RT}_{k}\n                  \n                and \n                  \n                     \n                        \n                           BDM\n                           k\n                        \n                     \n                     \n                     \\mathrm{BDM}_{k}\n                  \n                on triangles,\nComput. Math. Appl. 64 (2012), no. 8, 2765\u20132774."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_021","unstructured":"R. H. W. Hoppe and B. Wolmuth,\nHierarchical basis error estimators for Raviart\u2013Thomas discretizations of arbitrary order,\nSIAM J. Numer. Anal. 45 (2007), no. 3, 1123\u20131147."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_022","unstructured":"A. F. Ibrahimbegovic and A. G. Hughes,\nA mixed Raviart\u2013Thomas and Argyris element for the Biot\u2019s consolidation equations,\nInternat. J. Numer. Methods Engrg. 119 (2019), 1054\u20131074."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_023","doi-asserted-by":"crossref","unstructured":"M. K\u0159\u00ed\u017eek,\nOn the maximum angle condition for linear tetrahedral elements,\nSIAM J. Numer. Anal. 29 (1992), no. 2, 513\u2013520.","DOI":"10.1137\/0729031"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_024","unstructured":"Q. Lin and J. H. Pan,\nHigh accuracy for mixed finite element methods in Raviart\u2013Thomas element,\nJ. Comput. Math. 14 (1996), no. 2, 175\u2013182."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_025","doi-asserted-by":"crossref","unstructured":"P. Morin, R. H. Nochetto and K. G. Siebert,\nData oscillation and convergence of adaptive FEM,\nSIAM J. Numer. Anal. 38 (2000), no. 2, 466\u2013488.","DOI":"10.1137\/S0036142999360044"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_026","doi-asserted-by":"crossref","unstructured":"J.-C. N\u00e9d\u00e9lec,\nMixed finite elements in \n                  \n                     \n                        \n                           R\n                           3\n                        \n                     \n                     \n                     \\mathbf{R}^{3}\n                  \n               ,\nNumer. Math. 35 (1980), no. 3, 315\u2013341.","DOI":"10.1007\/BF01396415"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_027","doi-asserted-by":"crossref","unstructured":"P.-A. Raviart and J. M. Thomas,\nA mixed finite element method for 2nd order elliptic problems,\nMathematical Aspects of Finite Element Methods,\nLecture Notes in Math. 606,\nSpringer, Berlin (1977), 292\u2013315.","DOI":"10.1007\/BFb0064470"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_028","unstructured":"J. Roberts and J.-M. Thomas,\nMixed and hybrid finite element methods,\nResearch Report RR-0737, INRIA, 1987."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_029","doi-asserted-by":"crossref","unstructured":"M. E. Rognes, R. C. Kirby and A. Logg,\nEfficient assembly of \n                  \n                     \n                        \n                           H\n                           \u2062\n                           \n                              (\n                              div\n                              )\n                           \n                        \n                     \n                     \n                     H(\\mathrm{div})\n                  \n                and \n                  \n                     \n                        \n                           H\n                           \u2062\n                           \n                              (\n                              curl\n                              )\n                           \n                        \n                     \n                     \n                     H(\\mathrm{curl})\n                  \n                conforming finite elements,\nSIAM J. Sci. Comput. 31 (2009\/10), no. 6, 4130\u20134151.","DOI":"10.1137\/08073901X"},{"key":"2023100411150468233_j_cmam-2022-0235_ref_030","unstructured":"J.-M. Thomas,\nSur l\u2019analyse num\u00e9rique des m\u00e9thodes d\u2019\u00e9l\u00e9ments finis hybrides et mixtes,\nTh\u00e8se de doctorat d\u2019Etat, Universit\u00e9 Pierre et Marie Curie, Paris, 1977."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_031","unstructured":"M. Wang, Y. Zhang and B. Jin,\nA stabilized Raviart\u2013Thomas mixed finite element method for poromechanics with nonlinear elasticity,\nInt. J. Numer. Anal. Methods Geomech. 45 (2021), 2281\u20132300."},{"key":"2023100411150468233_j_cmam-2022-0235_ref_032","doi-asserted-by":"crossref","unstructured":"H. Yserentant,\nOn the multilevel splitting of finite element spaces,\nNumer. Math. 49 (1986), no. 4, 379\u2013412.","DOI":"10.1007\/BF01389538"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0235\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0235\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,4]],"date-time":"2023-10-04T11:16:12Z","timestamp":1696418172000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2022-0235\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,9,19]]},"references-count":32,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,4,1]]},"published-print":{"date-parts":[[2023,10,1]]}},"alternative-id":["10.1515\/cmam-2022-0235"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2022-0235","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2023,9,19]]}}}