{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,6,18]],"date-time":"2025-06-18T07:31:50Z","timestamp":1750231910963,"version":"3.40.5"},"reference-count":42,"publisher":"Walter de Gruyter GmbH","issue":"3","license":[{"start":{"date-parts":[[2023,7,25]],"date-time":"2023-07-25T00:00:00Z","timestamp":1690243200000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/100010663","name":"H2020 European Research Council","doi-asserted-by":"publisher","award":["865751","891734"],"award-info":[{"award-number":["865751","891734"]}],"id":[{"id":"10.13039\/100010663","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["PE 2143\/5-1"],"award-info":[{"award-number":["PE 2143\/5-1"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>This paper proposes novel computational multiscale methods for linear second-order elliptic partial differential equations in nondivergence form with heterogeneous coefficients satisfying a Cordes condition.\nThe construction follows the methodology of localized orthogonal decomposition (LOD) and provides operator-adapted coarse spaces by solving localized cell problems on a fine scale in the spirit of numerical homogenization.\nThe degrees of freedom of the coarse spaces are related to nonconforming and mixed finite element methods for homogeneous problems.\nThe rigorous error analysis of one exemplary approach shows that the favorable properties of the LOD methodology known from divergence-form PDEs, i.e., its applicability and accuracy beyond scale separation and periodicity, remain valid for problems in nondivergence form.<\/jats:p>","DOI":"10.1515\/cmam-2023-0040","type":"journal-article","created":{"date-parts":[[2023,7,24]],"date-time":"2023-07-24T13:42:28Z","timestamp":1690206148000},"page":"649-672","source":"Crossref","is-referenced-by-count":3,"title":["Computational Multiscale Methods for Nondivergence-Form Elliptic Partial Differential Equations"],"prefix":"10.1515","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-9838-6321","authenticated-orcid":false,"given":"Philip","family":"Freese","sequence":"first","affiliation":[{"name":"Institut f\u00fcr Mathematik , Technische Universit\u00e4t Hamburg , Am Schwarzenberg-Campus 3, 21073 Hamburg , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8954-4175","authenticated-orcid":false,"given":"Dietmar","family":"Gallistl","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Mathematik , Friedrich-Schiller-Universit\u00e4t Jena , Ernst-Abbe-Platz 2, 07743 Jena , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7213-556X","authenticated-orcid":false,"given":"Daniel","family":"Peterseim","sequence":"additional","affiliation":[{"name":"Institut f\u00fcr Mathematik & Centre for Advanced Analytics and Predictive Sciences (CAAPS) , Universit\u00e4t Augsburg , Universit\u00e4tsstr. 12a, 86159 Augsburg , Germany"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2934-8126","authenticated-orcid":false,"given":"Timo","family":"Sprekeler","sequence":"additional","affiliation":[{"name":"Department of Mathematics , National University of Singapore , 10 Lower Kent Ridge Road , Singapore 119076 , Singapore"}]}],"member":"374","published-online":{"date-parts":[[2023,7,25]]},"reference":[{"key":"2024070116104967691_j_cmam-2023-0040_ref_001","doi-asserted-by":"crossref","unstructured":"R. Altmann, P. Henning and D. Peterseim,\nNumerical homogenization beyond scale separation,\nActa Numer. 30 (2021), 1\u201386.","DOI":"10.1017\/S0962492921000015"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_002","doi-asserted-by":"crossref","unstructured":"D. Arjmand and G. Kreiss,\nAn equation-free approach for second order multiscale hyperbolic problems in non-divergence form,\nCommun. Math. Sci. 16 (2018), no. 8, 2317\u20132343.","DOI":"10.4310\/CMS.2018.v16.n8.a11"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_003","doi-asserted-by":"crossref","unstructured":"M. Avellaneda and F.-H. Lin,\nCompactness methods in the theory of homogenization. II. Equations in nondivergence form,\nComm. Pure Appl. Math. 42 (1989), no. 2, 139\u2013172.","DOI":"10.1002\/cpa.3160420203"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_004","doi-asserted-by":"crossref","unstructured":"I. Babuska and R. Lipton,\nOptimal local approximation spaces for generalized finite element methods with application to multiscale problems,\nMultiscale Model. Simul. 9 (2011), no. 1, 373\u2013406.","DOI":"10.1137\/100791051"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_005","doi-asserted-by":"crossref","unstructured":"A. Bensoussan, J.-L. Lions and G. Papanicolaou,\nAsymptotic Analysis for Periodic Structures,\nAMS Chelsea, Providence, 2011.","DOI":"10.1090\/chel\/374"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_006","doi-asserted-by":"crossref","unstructured":"G. Birkhoff and L. Mansfield,\nCompatible triangular finite elements,\nJ. Math. Anal. Appl. 47 (1974), 531\u2013553.","DOI":"10.1016\/0022-247X(74)90006-7"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_007","unstructured":"F. Bonizzoni, P. Freese and D. Peterseim,\nSuper-localized orthogonal decomposition for convection-dominated diffusion problems,\npreprint (2022), https:\/\/arxiv.org\/abs\/2206.01975."},{"key":"2024070116104967691_j_cmam-2023-0040_ref_008","doi-asserted-by":"crossref","unstructured":"F. Camilli and C. Marchi,\nRates of convergence in periodic homogenization of fully nonlinear uniformly elliptic PDEs,\nNonlinearity 22 (2009), no. 6, 1481\u20131498.","DOI":"10.1088\/0951-7715\/22\/6\/011"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_009","doi-asserted-by":"crossref","unstructured":"Y. Capdeboscq, T. Sprekeler and E. S\u00fcli,\nFinite element approximation of elliptic homogenization problems in nondivergence-form,\nESAIM Math. Model. Numer. Anal. 54 (2020), no. 4, 1221\u20131257.","DOI":"10.1051\/m2an\/2019093"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_010","doi-asserted-by":"crossref","unstructured":"P. G. Ciarlet,\nThe Finite Element Method for Elliptic Problems,\nStud. Math. Appl. 4,\nNorth-Holland, Amsterdam, 1978.","DOI":"10.1115\/1.3424474"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_011","doi-asserted-by":"crossref","unstructured":"Y. Efendiev, J. Galvis and T. Y. Hou,\nGeneralized multiscale finite element methods (GMsFEM),\nJ. Comput. Phys. 251 (2013), 116\u2013135.","DOI":"10.1016\/j.jcp.2013.04.045"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_012","doi-asserted-by":"crossref","unstructured":"C. Finlay and A. M. Oberman,\nApproximate homogenization of convex nonlinear elliptic PDEs,\nCommun. Math. Sci. 16 (2018), no. 7, 1895\u20131906.","DOI":"10.4310\/CMS.2018.v16.n7.a7"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_013","doi-asserted-by":"crossref","unstructured":"C. Finlay and A. M. Oberman,\nApproximate homogenization of fully nonlinear elliptic PDEs: Estimates and numerical results for Pucci type equations,\nJ. Sci. Comput. 77 (2018), no. 2, 936\u2013949.","DOI":"10.1007\/s10915-018-0730-x"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_014","unstructured":"P. Freese, M. Hauck and D. Peterseim,\nSuper-localized orthogonal decomposition for high-frequency Helmholtz problems, preprint (2021), https:\/\/arxiv.org\/abs\/2112.11368."},{"key":"2024070116104967691_j_cmam-2023-0040_ref_015","doi-asserted-by":"crossref","unstructured":"B. D. Froese and A. M. Oberman,\nNumerical averaging of non-divergence structure elliptic operators,\nCommun. Math. Sci. 7 (2009), no. 4, 785\u2013804.","DOI":"10.4310\/CMS.2009.v7.n4.a1"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_016","doi-asserted-by":"crossref","unstructured":"D. Gallistl,\nMorley finite element method for the eigenvalues of the biharmonic operator,\nIMA J. Numer. Anal. 35 (2015), no. 4, 1779\u20131811.","DOI":"10.1093\/imanum\/dru054"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_017","doi-asserted-by":"crossref","unstructured":"D. Gallistl,\nStable splitting of polyharmonic operators by generalized Stokes systems,\nMath. Comp. 86 (2017), no. 308, 2555\u20132577.","DOI":"10.1090\/mcom\/3208"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_018","doi-asserted-by":"crossref","unstructured":"D. Gallistl,\nVariational formulation and numerical analysis of linear elliptic equations in nondivergence form with Cordes coefficients,\nSIAM J. Numer. Anal. 55 (2017), no. 2, 737\u2013757.","DOI":"10.1137\/16M1080495"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_019","doi-asserted-by":"crossref","unstructured":"D. Gallistl,\nNumerical approximation of planar oblique derivative problems in nondivergence form,\nMath. Comp. 88 (2019), no. 317, 1091\u20131119.","DOI":"10.1090\/mcom\/3371"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_020","doi-asserted-by":"crossref","unstructured":"D. Gallistl, T. Sprekeler and E. S\u00fcli,\nMixed finite element approximation of periodic Hamilton\u2013Jacobi\u2013Bellman problems with application to numerical homogenization,\nMultiscale Model. Simul. 19 (2021), no. 2, 1041\u20131065.","DOI":"10.1137\/20M1371397"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_021","unstructured":"X. Guo, T. Sprekeler and H. V. Tran,\nCharacterizations of diffusion matrices in homogenization of elliptic equations in nondivergence-form,\npreprint (2022), https:\/\/arxiv.org\/abs\/2201.01974."},{"key":"2024070116104967691_j_cmam-2023-0040_ref_022","doi-asserted-by":"crossref","unstructured":"X. Guo, H. V. Tran and Y. Yu,\nRemarks on optimal rates of convergence in periodic homogenization of linear elliptic equations in non-divergence form,\nPartial Differ. Equ. Appl. 1 (2020), no. 4, Paper No. 15.","DOI":"10.1007\/s42985-020-00017-z"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_023","doi-asserted-by":"crossref","unstructured":"M. Hauck and D. Peterseim,\nMulti-resolution localized orthogonal decomposition for Helmholtz problems,\nMultiscale Model. Simul. 20 (2022), no. 2, 657\u2013684.","DOI":"10.1137\/21M1414607"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_024","doi-asserted-by":"crossref","unstructured":"M. Hauck and D. Peterseim,\nSuper-localization of elliptic multiscale problems,\nMath. Comp. 92 (2023), no. 341, 981\u20131003.","DOI":"10.1090\/mcom\/3798"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_025","doi-asserted-by":"crossref","unstructured":"P. Henning and D. Peterseim,\nOversampling for the multiscale finite element method,\nMultiscale Model. Simul. 11 (2013), no. 4, 1149\u20131175.","DOI":"10.1137\/120900332"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_026","doi-asserted-by":"crossref","unstructured":"V. V. Jikov, S. M. Kozlov and O. A. Ole\u012dnik,\nHomogenization of Differential Operators and Integral Functionals,\nSpringer, Berlin, 1994.","DOI":"10.1007\/978-3-642-84659-5"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_027","doi-asserted-by":"crossref","unstructured":"E. L. Kawecki and T. Sprekeler,\nDiscontinuous Galerkin and \n                  \n                     \n                        \n                           C\n                           0\n                        \n                     \n                     \n                     C^{0}\n                  \n               -IP finite element approximation of periodic Hamilton\u2013Jacobi\u2013Bellman\u2013Isaacs problems with application to numerical homogenization,\nESAIM Math. Model. Numer. Anal. 56 (2022), no. 2, 679\u2013704.","DOI":"10.1051\/m2an\/2022017"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_028","doi-asserted-by":"crossref","unstructured":"S. Kim and K.-A. Lee,\nHigher order convergence rates in theory of homogenization: Equations of non-divergence form,\nArch. Ration. Mech. Anal. 219 (2016), no. 3, 1273\u20131304.","DOI":"10.1007\/s00205-015-0921-7"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_029","doi-asserted-by":"crossref","unstructured":"R. Kornhuber, D. Peterseim and H. Yserentant,\nAn analysis of a class of variational multiscale methods based on subspace decomposition,\nMath. Comp. 87 (2018), no. 314, 2765\u20132774.","DOI":"10.1090\/mcom\/3302"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_030","doi-asserted-by":"crossref","unstructured":"C. Ma, R. Scheichl and T. Dodwell,\nNovel design and analysis of generalized finite element methods based on locally optimal spectral approximations,\nSIAM J. Numer. Anal. 60 (2022), no. 1, 244\u2013273.","DOI":"10.1137\/21M1406179"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_031","doi-asserted-by":"crossref","unstructured":"A. M\u00e5lqvist and D. Peterseim,\nLocalization of elliptic multiscale problems,\nMath. Comp. 83 (2014), no. 290, 2583\u20132603.","DOI":"10.1090\/S0025-5718-2014-02868-8"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_032","doi-asserted-by":"crossref","unstructured":"A. M\u00e5lqvist and D. Peterseim,\nNumerical Homogenization by Localized Orthogonal Decomposition,\nSIAM Spotlights 5,\nSociety for Industrial and Applied Mathematics, Philadelphia,2021.","DOI":"10.1137\/1.9781611976458"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_033","doi-asserted-by":"crossref","unstructured":"R. Maier,\nA high-order approach to elliptic multiscale problems with general unstructured coefficients,\nSIAM J. Numer. Anal. 59 (2021), no. 2, 1067\u20131089.","DOI":"10.1137\/20M1364321"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_034","doi-asserted-by":"crossref","unstructured":"H. Owhadi,\nMultigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games,\nSIAM Rev. 59 (2017), no. 1, 99\u2013149.","DOI":"10.1137\/15M1013894"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_035","doi-asserted-by":"crossref","unstructured":"H. Owhadi and L. Zhang,\nMetric-based upscaling,\nComm. Pure Appl. Math. 60 (2007), no. 5, 675\u2013723.","DOI":"10.1002\/cpa.20163"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_036","doi-asserted-by":"crossref","unstructured":"H. Owhadi, L. Zhang and L. Berlyand,\nPolyharmonic homogenization, rough polyharmonic splines and sparse super-localization,\nESAIM Math. Model. Numer. Anal. 48 (2014), no. 2, 517\u2013552.","DOI":"10.1051\/m2an\/2013118"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_037","doi-asserted-by":"crossref","unstructured":"I. Smears and E. S\u00fcli,\nDiscontinuous Galerkin finite element approximation of nondivergence form elliptic equations with Cord\u00e8s coefficients,\nSIAM J. Numer. Anal. 51 (2013), no. 4, 2088\u20132106.","DOI":"10.1137\/120899613"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_038","doi-asserted-by":"crossref","unstructured":"I. Smears and E. S\u00fcli,\nDiscontinuous Galerkin finite element approximation of Hamilton\u2013Jacobi\u2013Bellman equations with Cordes coefficients,\nSIAM J. Numer. Anal. 52 (2014), no. 2, 993\u20131016.","DOI":"10.1137\/130909536"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_039","unstructured":"T. Sprekeler,\nHomogenization of nondivergence-form elliptic equations with discontinuous coefficients and finite element approximation of the homogenized problem, preprint (2023), https:\/\/arxiv.org\/abs\/2305.19833."},{"key":"2024070116104967691_j_cmam-2023-0040_ref_040","doi-asserted-by":"crossref","unstructured":"T. Sprekeler and H. V. Tran,\nOptimal convergence rates for elliptic homogenization problems in nondivergence-form: Analysis and numerical illustrations,\nMultiscale Model. Simul. 19 (2021), no. 3, 1453\u20131473.","DOI":"10.1137\/20M137121X"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_041","doi-asserted-by":"crossref","unstructured":"D. B. Szyld,\nThe many proofs of an identity on the norm of oblique projections,\nNumer. Algorithms 42 (2006), no. 3\u20134, 309\u2013323.","DOI":"10.1007\/s11075-006-9046-2"},{"key":"2024070116104967691_j_cmam-2023-0040_ref_042","doi-asserted-by":"crossref","unstructured":"M. Wang and J. Xu,\nMinimal finite element spaces for \n                  \n                     \n                        \n                           2\n                           \u2062\n                           m\n                        \n                     \n                     \n                     2m\n                  \n               -th-order partial differential equations in \n                  \n                     \n                        \n                           R\n                           n\n                        \n                     \n                     \n                     R^{n}\n                  \n               ,\nMath. Comp. 82 (2013), no. 281, 25\u201343.","DOI":"10.1090\/S0025-5718-2012-02611-1"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0040\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0040\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,7,1]],"date-time":"2024-07-01T16:13:51Z","timestamp":1719850431000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyter.com\/document\/doi\/10.1515\/cmam-2023-0040\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,7,25]]},"references-count":42,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2024,6,26]]},"published-print":{"date-parts":[[2024,7,1]]}},"alternative-id":["10.1515\/cmam-2023-0040"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2023-0040","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"type":"print","value":"1609-4840"},{"type":"electronic","value":"1609-9389"}],"subject":[],"published":{"date-parts":[[2023,7,25]]}}}