{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T04:13:31Z","timestamp":1751516011313,"version":"3.41.0"},"reference-count":40,"publisher":"Walter de Gruyter GmbH","issue":"3","funder":[{"DOI":"10.13039\/501100001659","name":"Deutsche Forschungsgemeinschaft","doi-asserted-by":"publisher","award":["CA 1159\/1-4","PE 2143\/1-6"],"award-info":[{"award-number":["CA 1159\/1-4","PE 2143\/1-6"]}],"id":[{"id":"10.13039\/501100001659","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["865751"],"award-info":[{"award-number":["865751"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2025,7,1]]},"abstract":"<jats:title>Abstract<\/jats:title>\n               <jats:p>We present the Super-Localized Orthogonal Decomposition (SLOD) method for the numerical homogenization of linear elasticity problems with multiscale microstructures modeled by a heterogeneous coefficient field without any periodicity or scale separation assumptions. Compared to the established Localized Orthogonal Decomposition (LOD) and its linear localization approach, SLOD achieves significantly improved sparsity properties through a nonlinear superlocalization technique, leading to computationally efficient solutions with significantly less oversampling \u2013 without compromising accuracy. We generalize the method to vector-valued problems and provide a supporting numerical analysis. We also present a scalable implementation of SLOD using the deal.II finite element library, demonstrating its feasibility for high-performance simulations. Initial numerical experiments indicate the potential of SLOD for addressing computational challenges in multiscale elasticity.<\/jats:p>","DOI":"10.1515\/cmam-2025-0005","type":"journal-article","created":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T17:06:46Z","timestamp":1751476006000},"page":"561-579","source":"Crossref","is-referenced-by-count":0,"title":["Super-Localized Orthogonal Decomposition Method for Heterogeneous Linear Elasticity"],"prefix":"10.1515","volume":"25","author":[{"given":"Camilla","family":"Belponer","sequence":"first","affiliation":[{"name":"Institute of Mathematics , 26522 University of Augsburg , Universit\u00e4tsstr. 12a, 86159 Augsburg , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jos\u00e9 C.","family":"Garay","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , 26522 University of Augsburg , Universit\u00e4tsstr. 12a, 86159 Augsburg , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Peter","family":"Munch","sequence":"additional","affiliation":[{"name":"Institute of Mathematics , Technische Universit\u00e4t Berlin , Stra\u00dfe des 17. Juni 136, 10623 Berlin , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Daniel","family":"Peterseim","sequence":"additional","affiliation":[{"name":"Institute of Mathematics ; and Centre for Advanced Analytics and Predictive Sciences (CAAPS) , 26522 University of Augsburg , Universit\u00e4tsstr. 12a, 86159 Augsburg , Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"374","published-online":{"date-parts":[[2025,6,19]]},"reference":[{"key":"2025070217063908509_j_cmam-2025-0005_ref_001","doi-asserted-by":"crossref","unstructured":"P. C.  Africa, D.  Arndt, W.  Bangerth, B.  Blais, M.  Fehling, R.  Gassm\u00f6ller, T.  Heister, L.  Heltai, S.  Kinnewig, M.  Kronbichler, M.  Maier, P.  Munch, M.  Schreter-Fleischhacker, J. P.  Thiele, B.  Turcksin, D.  Wells and V.  Yushutin,\nThe deal.ii library, version 9.6,\nJ. Numer. Math. 32 (2024), no. 4, 369\u2013380.","DOI":"10.1515\/jnma-2024-0137"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_002","doi-asserted-by":"crossref","unstructured":"R.  Altmann, E.  Chung, R.  Maier, D.  Peterseim and S.-M.  Pun,\nComputational multiscale methods for linear heterogeneous poroelasticity,\nJ. Comput. Math. 38 (2020), no. 1, 41\u201357.","DOI":"10.4208\/jcm.1902-m2018-0186"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_003","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":"2025070217063908509_j_cmam-2025-0005_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":"2025070217063908509_j_cmam-2025-0005_ref_005","doi-asserted-by":"crossref","unstructured":"I.  Babu\u0161ka, R.  Lipton, P.  Sinz and M.  Stuebner,\nMultiscale-spectral GFEM and optimal oversampling,\nComput. Methods Appl. Mech. Engrg. 364 (2020), Article ID 112960.","DOI":"10.1016\/j.cma.2020.112960"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_006","doi-asserted-by":"crossref","unstructured":"X.  Blanc and C.  Le Bris,\nHomogenization Theory for Multiscale Problems,\nMS&A. Model. Simul. Appl. 21,\nSpringer, Cham, 2023.","DOI":"10.1007\/978-3-031-21833-0"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_007","doi-asserted-by":"crossref","unstructured":"F.  Bonizzoni, P.  Freese and D.  Peterseim,\nSuper-localized orthogonal decomposition for convection-dominated diffusion problems,\nBIT 64 (2024), no. 3, Paper No. 33.","DOI":"10.1007\/s10543-024-01035-8"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_008","doi-asserted-by":"crossref","unstructured":"F.  Bonizzoni, M.  Hauck and D.  Peterseim,\nA reduced basis super-localized orthogonal decomposition for reaction-convection-diffusion problems,\nJ. Comput. Phys. 499 (2024), Article ID 112698.","DOI":"10.1016\/j.jcp.2023.112698"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_009","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner, J. C.  Garay and L.-Y.  Sung,\nAdditive Schwarz preconditioners for a localized orthogonal decomposition method,\nElectron. Trans. Numer. Anal. 54 (2021), 234\u2013255.","DOI":"10.1553\/etna_vol54s234"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_010","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner, J. C.  Garay and L.-Y.  Sung,\nMultiscale finite element methods for an elliptic optimal control problem with rough coefficients,\nJ. Sci. Comput. 91 (2022), no. 3, Paper No. 76.","DOI":"10.1007\/s10915-022-01834-7"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_011","doi-asserted-by":"crossref","unstructured":"S. C.  Brenner and L. R.  Scott,\nThe Mathematical Theory of Finite Element Methods, 3rd ed.,\nTexts Appl. Math. 15,\nSpringer, New York, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_012","doi-asserted-by":"crossref","unstructured":"E.  Chung, Y.  Efendiev and T. Y.  Hou,\nMultiscale Model Reduction \u2013 Multiscale Finite Element Methods and Their Generalizations,\nAppl. Math. Sci. 212,\nSpringer, Cham, 2023.","DOI":"10.1007\/978-3-031-20409-8"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_013","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":"2025070217063908509_j_cmam-2025-0005_ref_014","unstructured":"Y.  Efendiev and T. Y.  Hou,\nMultiscale Finite Element Methods,\nSurv. Tutor. Appl. Math. Sci. 4,\nSpringer, New York, 2009."},{"key":"2025070217063908509_j_cmam-2025-0005_ref_015","doi-asserted-by":"crossref","unstructured":"P.  Freese, M.  Hauck, T.  Keil and D.  Peterseim,\nA super-localized generalized finite element method,\nNumer. Math. 156 (2024), no. 1, 205\u2013235.","DOI":"10.1007\/s00211-023-01386-4"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_016","doi-asserted-by":"crossref","unstructured":"P.  Freese, M.  Hauck and D.  Peterseim,\nSuper-localized orthogonal decomposition for high-frequency Helmholtz problems,\nSIAM J. Sci. Comput. 46 (2024), no. 4, A2377\u2013A2397.","DOI":"10.1137\/21M1465950"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_017","doi-asserted-by":"crossref","unstructured":"S.  Fu, R.  Altmann, E. T.  Chung, R.  Maier, D.  Peterseim and S.-M.  Pun,\nComputational multiscale methods for linear poroelasticity with high contrast,\nJ. Comput. Phys. 395 (2019), 286\u2013297.","DOI":"10.1016\/j.jcp.2019.06.027"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_018","unstructured":"J. C.  Garay, H.  Mohr, D.  Peterseim and C.  Zimmer,\nHierarchical super-localized orthogonal decomposition method,\npreprint (2024), https:\/\/arxiv.org\/abs\/2407.18671."},{"key":"2025070217063908509_j_cmam-2025-0005_ref_019","unstructured":"M.  Hauck and A.  Lozinski,\nA localized orthogonal decomposition method for heterogeneous stokes problems,\npreprint (2024), https:\/\/arxiv.org\/abs\/2410.14514."},{"key":"2025070217063908509_j_cmam-2025-0005_ref_020","doi-asserted-by":"crossref","unstructured":"M.  Hauck and A.  M\u00e5lqvist,\nSuper-localization of spatial network models,\nNumer. Math. 156 (2024), no. 3, 901\u2013926.","DOI":"10.1007\/s00211-024-01410-1"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_021","doi-asserted-by":"crossref","unstructured":"M.  Hauck, H.  Mohr and D.  Peterseim,\nA simple collocation-type approach to numerical stochastic homogenization,\nMultiscale Model. Simul. 23 (2025), 374\u2013396.","DOI":"10.1137\/24M1650107"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_022","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":"2025070217063908509_j_cmam-2025-0005_ref_023","doi-asserted-by":"crossref","unstructured":"P.  Henning and A.  Persson,\nA multiscale method for linear elasticity reducing Poisson locking,\nComput. Methods Appl. Mech. Engrg. 310 (2016), 156\u2013171.","DOI":"10.1016\/j.cma.2016.06.034"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_024","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":"2025070217063908509_j_cmam-2025-0005_ref_025","doi-asserted-by":"crossref","unstructured":"T. Y.  Hou and X.-H.  Wu,\nA multiscale finite element method for elliptic problems in composite materials and porous media,\nJ. Comput. Phys. 134 (1997), no. 1, 169\u2013189.","DOI":"10.1006\/jcph.1997.5682"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_026","doi-asserted-by":"crossref","unstructured":"A.  Klawonn, M.  K\u00fchn and O.  Rheinbach,\nAdaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems,\nElectron. Trans. Numer. Anal. 49 (2018), 1\u201327.","DOI":"10.1553\/etna_vol49s1"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_027","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":"2025070217063908509_j_cmam-2025-0005_ref_028","doi-asserted-by":"crossref","unstructured":"C.  Ma,\nA unified framework for multiscale spectral generalized fems and low-rank approximations to multiscale pdes,\nFound. Comput. Math. (2025), 10.1007\/s10208-025-09711-z.","DOI":"10.1007\/s10208-025-09711-z"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_029","doi-asserted-by":"crossref","unstructured":"A. L.  Madureira and M.  Sarkis,\nHybrid localized spectral decomposition for multiscale problems,\nSIAM J. Numer. Anal. 59 (2021), no. 2, 829\u2013863.","DOI":"10.1137\/20M1314896"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_030","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":"2025070217063908509_j_cmam-2025-0005_ref_031","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":"2025070217063908509_j_cmam-2025-0005_ref_032","doi-asserted-by":"crossref","unstructured":"A.  M\u00e5lqvist and A.  Persson,\nA generalized finite element method for linear thermoelasticity,\nESAIM Math. Model. Numer. Anal. 51 (2017), no. 4, 1145\u20131171.","DOI":"10.1051\/m2an\/2016054"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_033","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":"2025070217063908509_j_cmam-2025-0005_ref_034","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":"2025070217063908509_j_cmam-2025-0005_ref_035","doi-asserted-by":"crossref","unstructured":"H.  Owhadi,\nBayesian numerical homogenization,\nMultiscale Model. Simul. 13 (2015), no. 3, 812\u2013828.","DOI":"10.1137\/140974596"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_036","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":"2025070217063908509_j_cmam-2025-0005_ref_037","doi-asserted-by":"crossref","unstructured":"H.  Owhadi and C.  Scovel,\nOperator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization,\nCambridge Monogr. Appl. Comput. Math. 35,\nCambridge University, Cambridge, 2019.","DOI":"10.1017\/9781108594967"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_038","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":"2025070217063908509_j_cmam-2025-0005_ref_039","doi-asserted-by":"crossref","unstructured":"D.  Peterseim and R.  Scheichl,\nRobust numerical upscaling of elliptic multiscale problems at high contrast,\nComput. Methods Appl. Math. 16 (2016), no. 4, 579\u2013603.","DOI":"10.1515\/cmam-2016-0022"},{"key":"2025070217063908509_j_cmam-2025-0005_ref_040","doi-asserted-by":"crossref","unstructured":"D.  Peterseim, J.  W\u00e4rneg\u00e5rd and C.  Zimmer,\nSuper-localised wave function approximation of Bose\u2013Einstein condensates,\nJ. Comput. Phys. 510 (2024), Article ID 113097.","DOI":"10.1016\/j.jcp.2024.113097"}],"container-title":["Computational Methods in Applied Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2025-0005\/xml","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2025-0005\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,2]],"date-time":"2025-07-02T17:07:36Z","timestamp":1751476056000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.degruyterbrill.com\/document\/doi\/10.1515\/cmam-2025-0005\/html"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,19]]},"references-count":40,"journal-issue":{"issue":"3","published-online":{"date-parts":[[2025,6,17]]},"published-print":{"date-parts":[[2025,7,1]]}},"alternative-id":["10.1515\/cmam-2025-0005"],"URL":"https:\/\/doi.org\/10.1515\/cmam-2025-0005","relation":{},"ISSN":["1609-4840","1609-9389"],"issn-type":[{"value":"1609-4840","type":"print"},{"value":"1609-9389","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,19]]}}}