{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,15]],"date-time":"2026-07-15T18:47:53Z","timestamp":1784141273046,"version":"3.55.0"},"reference-count":98,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T00:00:00Z","timestamp":1745884800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T00:00:00Z","timestamp":1745884800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2026,6]]},"DOI":"10.1007\/s10208-025-09711-z","type":"journal-article","created":{"date-parts":[[2025,4,29]],"date-time":"2025-04-29T11:34:02Z","timestamp":1745926442000},"page":"1699-1758","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A Unified Framework for Multiscale Spectral Generalized FEMs and Low-Rank Approximations to Multiscale PDEs"],"prefix":"10.1007","volume":"26","author":[{"given":"Chupeng","family":"Ma","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2025,4,29]]},"reference":[{"issue":"04","key":"9711_CR1","doi-asserted-by":"publisher","first-page":"615","DOI":"10.1142\/S0218202506001285","volume":"16","author":"Assyr Abdulle","year":"2006","unstructured":"Assyr Abdulle, Analysis of a heterogeneous multiscale FEM for problems in elasticity, Math. Models Methods Appl. Sci. 16 (2006), no.\u00a004, 615\u2013635.","journal-title":"Math. Models Methods Appl. Sci."},{"issue":"4","key":"9711_CR2","doi-asserted-by":"publisher","first-page":"589","DOI":"10.1007\/s00211-013-0578-9","volume":"126","author":"Assyr Abdulle","year":"2014","unstructured":"Assyr Abdulle and Martin\u00a0E Huber, Discontinuous galerkin finite element heterogeneous multiscale method for advection\u2013diffusion problems with multiple scales, Numer. Math. 126 (2014), no.\u00a04, 589\u2013633.","journal-title":"Numer. Math."},{"key":"9711_CR3","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492912000025","volume":"21","author":"Assyr Abdulle","year":"2012","unstructured":"Assyr Abdulle, E\u00a0Weinan, Bj\u00f6rn Engquist, and Eric Vanden-Eijnden, The heterogeneous multiscale method, Acta Numerica 21 (2012), 1\u201387.","journal-title":"Acta Numerica"},{"key":"9711_CR4","doi-asserted-by":"crossref","unstructured":"Christian Alber, Chupeng Ma, and Robert Scheichl, A mixed multiscale spectral generalized finite element method, Numer. Math. (2025), 1\u201340.","DOI":"10.1007\/s00211-024-01446-3"},{"key":"9711_CR5","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1017\/S0962492921000015","volume":"30","author":"Robert Altmann","year":"2021","unstructured":"Robert Altmann, Patrick Henning, and Daniel Peterseim, Numerical homogenization beyond scale separation, Acta Numerica 30 (2021), 1\u201386.","journal-title":"Acta Numerica"},{"issue":"3","key":"9711_CR6","doi-asserted-by":"publisher","first-page":"A1451","DOI":"10.1137\/120903476","volume":"37","author":"Patrick Amestoy","year":"2015","unstructured":"Patrick Amestoy, Cleve Ashcraft, Olivier Boiteau, Alfredo Buttari, Jean-Yves L\u2019Excellent, and Cl\u00e9ment Weisbecker, Improving multifrontal methods by means of block low-rank representations, SIAM J. Sci. Comput. 37 (2015), no.\u00a03, A1451\u2013A1474.","journal-title":"SIAM J. Sci. Comput."},{"key":"9711_CR7","doi-asserted-by":"publisher","first-page":"21","DOI":"10.1016\/j.camwa.2022.11.011","volume":"130","author":"Niklas Angleitner","year":"2023","unstructured":"Niklas Angleitner, Markus Faustmann, and Jens\u00a0Markus Melenk, Exponential meshes and H-matrices, Comput. Math. Appl. 130 (2023), 21\u201340.","journal-title":"Comput. Math. Appl."},{"key":"9711_CR8","doi-asserted-by":"crossref","unstructured":"Ivo Babuska, Survey lectures on the mathematical foundations of the finite element method, The Mathematical Foundations of the Finite Element Method with Applicaions to Partial Differential Equations (1972), 3\u2013359.","DOI":"10.1016\/B978-0-12-068650-6.50006-X"},{"issue":"2","key":"9711_CR9","doi-asserted-by":"publisher","first-page":"493","DOI":"10.1051\/m2an\/2013117","volume":"48","author":"Ivo Babu\u0161ka","year":"2014","unstructured":"Ivo Babu\u0161ka, Xu\u00a0Huang, and Robert Lipton, Machine computation using the exponentially convergent multiscale spectral generalized finite element method, ESAIM: Math. Model. Numer. Anal. 48 (2014), no.\u00a02, 493\u2013515.","journal-title":"ESAIM: Math. Model. Numer. Anal."},{"issue":"1","key":"9711_CR10","doi-asserted-by":"publisher","first-page":"373","DOI":"10.1137\/100791051","volume":"9","author":"Ivo Babu\u0161ka","year":"2011","unstructured":"Ivo Babu\u0161ka and Robert Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul. 9 (2011), no.\u00a01, 373\u2013406.","journal-title":"Multiscale Model. Simul."},{"key":"9711_CR11","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2020.112960","volume":"364","author":"Ivo Babu\u0161ka","year":"2020","unstructured":"Ivo Babu\u0161ka, Robert Lipton, Paul Sinz, and Michael Stuebner, Multiscale-Spectral GFEM and optimal oversampling, Comput. Methods Appl. Mech. Eng. 364 (2020), 112960.","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"4","key":"9711_CR12","doi-asserted-by":"publisher","first-page":"727","DOI":"10.1002\/(SICI)1097-0207(19970228)40:4<727::AID-NME86>3.0.CO;2-N","volume":"40","author":"Ivo Babu\u0161ka","year":"1997","unstructured":"Ivo Babu\u0161ka and Jens\u00a0M Melenk, The partition of unity method, Int. J. Numer. Methods Eng. 40 (1997), no.\u00a04, 727\u2013758.","journal-title":"Int. J. Numer. Methods Eng."},{"issue":"230","key":"9711_CR13","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1090\/S0025-5718-99-01085-6","volume":"69","author":"Ivo Babu\u0161ka","year":"2000","unstructured":"Ivo Babu\u0161ka and John Osborn, Can a finite element method perform arbitrarily badly?, Mathematics of computation 69 (2000), no.\u00a0230, 443\u2013462.","journal-title":"Mathematics of computation"},{"issue":"304","key":"9711_CR14","doi-asserted-by":"publisher","first-page":"701","DOI":"10.1090\/mcom\/3132","volume":"86","author":"Markus Bachmayr","year":"2017","unstructured":"Markus Bachmayr and Albert Cohen, Kolmogorov widths and low-rank approximations of parametric elliptic pdes, Math. Comp. 86 (2017), no.\u00a0304, 701\u2013724.","journal-title":"Math. Comp."},{"issue":"4","key":"9711_CR15","doi-asserted-by":"publisher","first-page":"1661","DOI":"10.1093\/imanum\/drx052","volume":"38","author":"Markus Bachmayr","year":"2018","unstructured":"Markus Bachmayr, Albert Cohen, and Wolfgang Dahmen, Parametric pdes: sparse or low-rank approximations?, IMA J. Numer. Anal. 38 (2018), no.\u00a04, 1661\u20131708.","journal-title":"IMA J. Numer. Anal."},{"issue":"251","key":"9711_CR16","doi-asserted-by":"publisher","first-page":"1179","DOI":"10.1090\/S0025-5718-04-01716-8","volume":"74","author":"Mario Bebendorf","year":"2005","unstructured":"Mario Bebendorf, Efficient inversion of the galerkin matrix of general second-order elliptic operators with nonsmooth coefficients, Math. Comp. 74 (2005), no.\u00a0251, 1179\u20131199.","journal-title":"Math. Comp."},{"key":"9711_CR17","doi-asserted-by":"crossref","unstructured":"Mario Bebendorf and Wolfgang Hackbusch, Existence of$$\\cal{H}$$-matrix approximants to the inverse FE-matrix of elliptic operators with$${L}^{\\infty }$$-coefficients, Numer. Math. 95 (2003), 1\u201328.","DOI":"10.1007\/s00211-002-0445-6"},{"key":"9711_CR18","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2024.113013","volume":"508","author":"Jean B\u00e9n\u00e9zech","year":"2024","unstructured":"Jean B\u00e9n\u00e9zech, Linus Seelinger, Peter Bastian, Richard Butler, Timothy Dodwell, Chupeng Ma, and Robert Scheichl, Scalable multiscale-spectral GFEM with an application to composite aero-structures, J. Comput. Phys. 508 (2024), 113013.","journal-title":"J. Comput. Phys."},{"key":"9711_CR19","doi-asserted-by":"crossref","unstructured":"Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, vol. 374, American Mathematical Soc., 2011.","DOI":"10.1090\/chel\/374"},{"key":"9711_CR20","unstructured":"Serge Bernstein, Sur l\u2019ordre de la meilleure approximation des fonctions continues par des polyn\u00f4mes de degr\u00e9 donn\u00e9, vol.\u00a04, Hayez, imprimeur des acad\u00e9mies royales, 1912."},{"key":"9711_CR21","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-642-36519-5","volume-title":"Mixed finite element methods and applications","author":"Daniele Boffi","year":"2013","unstructured":"Daniele Boffi, Franco Brezzi, and Michel Fortin, Mixed finite element methods and applications, Springer-Verlag, Berlin, 2013."},{"issue":"3","key":"9711_CR22","doi-asserted-by":"publisher","first-page":"33","DOI":"10.1007\/s10543-024-01035-8","volume":"64","author":"Francesca Bonizzoni","year":"2024","unstructured":"Francesca Bonizzoni, Philip Freese, and Daniel Peterseim, Super-localized orthogonal decomposition for convection-dominated diffusion problems, BIT Numer. Math. 64 (2024), no.\u00a03, 33.","journal-title":"BIT Numer. Math."},{"key":"9711_CR23","doi-asserted-by":"crossref","unstructured":"Steffen B\u00f6rm, Approximation of solution operators of elliptic partial differential equations by$$\\cal{H}$$- and$$\\cal{H}^{2}$$-matrices, Numer. Math. 115 (2010), no.\u00a02, 165\u2013193.","DOI":"10.1007\/s00211-009-0278-7"},{"key":"9711_CR24","doi-asserted-by":"crossref","unstructured":"Steffen B\u00f6rm, Efficient numerical methods for non-local operators: H2-matrix compression, algorithms and analysis, vol.\u00a014, European Mathematical Society, 2010.","DOI":"10.4171\/091"},{"issue":"3","key":"9711_CR25","doi-asserted-by":"publisher","first-page":"409","DOI":"10.1007\/s00211-015-0757-y","volume":"133","author":"Steffen B\u00f6rm","year":"2016","unstructured":"Steffen B\u00f6rm and Sven Christophersen, Approximation of integral operators by green quadrature and nested cross approximation, Numer. Math. 133 (2016), no.\u00a03, 409\u2013442.","journal-title":"Numer. Math."},{"issue":"3","key":"9711_CR26","doi-asserted-by":"publisher","first-page":"567","DOI":"10.1007\/s11075-012-9679-2","volume":"64","author":"Steffen B\u00f6rm","year":"2013","unstructured":"Steffen B\u00f6rm and Jessica G\u00f6rdes, Low-rank approximation of integral operators by using the green formula and quadrature, Numer. Algorithms 64 (2013), no.\u00a03, 567\u2013592.","journal-title":"Numer. Algorithms"},{"issue":"39","key":"9711_CR27","doi-asserted-by":"publisher","first-page":"e2303904120","DOI":"10.1073\/pnas.2303904120","volume":"120","author":"Nicolas Boull\u00e9","year":"2023","unstructured":"Nicolas Boull\u00e9, Diana Halikias, and Alex Townsend, Elliptic PDE learning is provably data-efficient, Proc. Natl. Acad. Sci. 120 (2023), no.\u00a039, e2303904120.","journal-title":"Proc. Natl. Acad. Sci."},{"key":"9711_CR28","doi-asserted-by":"crossref","unstructured":"Nicolas Boull\u00e9, Seick Kim, Tianyi Shi, and Alex Townsend, Learning Green\u2019s functions associated with parabolic partial differential equations, J. Mach. Learn. Res. 23 (2022), no.\u00a0218.","DOI":"10.1007\/s10208-022-09556-w"},{"issue":"2","key":"9711_CR29","doi-asserted-by":"publisher","first-page":"709","DOI":"10.1007\/s10208-022-09556-w","volume":"23","author":"Nicolas Boull\u00e9","year":"2023","unstructured":"Nicolas Boull\u00e9 and Alex Townsend, Learning elliptic partial differential equations with randomized linear algebra, Found. Comput. Math. 23 (2023), no.\u00a02, 709\u2013739.","journal-title":"Found. Comput. Math."},{"key":"9711_CR30","doi-asserted-by":"crossref","unstructured":"Dietrich Braess, Finite elements: Theory, fast solvers, and applications in solid mechanics, Cambridge University Press, 2007.","DOI":"10.1017\/CBO9780511618635"},{"key":"9711_CR31","doi-asserted-by":"crossref","unstructured":"Susanne\u00a0C Brenner, L\u00a0Ridgway Scott, and L\u00a0Ridgway Scott, The mathematical theory of finite element methods, vol.\u00a03, Springer, 2008.","DOI":"10.1007\/978-0-387-75934-0"},{"issue":"4","key":"9711_CR32","doi-asserted-by":"publisher","first-page":"A2120","DOI":"10.1137\/17M1138480","volume":"40","author":"Andreas Buhr","year":"2018","unstructured":"Andreas Buhr and Kathrin Smetana, Randomized local model order reduction, SIAM J. Sci. Comput. 40 (2018), no.\u00a04, A2120\u2013A2151.","journal-title":"SIAM J. Sci. Comput."},{"key":"9711_CR33","doi-asserted-by":"crossref","unstructured":"Victor\u00a0M Calo, Eric\u00a0T Chung, Yalchin Efendiev, and Wing\u00a0Tat Leung, Multiscale stabilization for convection-dominated diffusion in heterogeneous media, Comput. Methods Appl. Mech. Eng. 304 (2016), 359\u2013377.","DOI":"10.1016\/j.cma.2016.02.014"},{"issue":"4","key":"9711_CR34","doi-asserted-by":"publisher","first-page":"1727","DOI":"10.1137\/080734339","volume":"7","author":"Emmanuel Candes","year":"2009","unstructured":"Emmanuel Candes, Laurent Demanet, and Lexing Ying, A fast butterfly algorithm for the computation of Fourier integral operators, Multiscale Model. Simul. 7 (2009), no.\u00a04, 1727\u20131750.","journal-title":"Multiscale Model. Simul."},{"key":"9711_CR35","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1016\/j.jcp.2017.09.013","volume":"351","author":"St\u00e9phanie Chaillat","year":"2017","unstructured":"St\u00e9phanie Chaillat, Luca Desiderio, and Patrick Ciarlet, Theory and implementation of H-matrix based iterative and direct solvers for helmholtz and elastodynamic oscillatory kernels, J. Comput. Phys. 351 (2017), 165\u2013186.","journal-title":"J. Comput. Phys."},{"issue":"5","key":"9711_CR36","doi-asserted-by":"publisher","first-page":"2261","DOI":"10.1137\/090775932","volume":"31","author":"Shiv Chandrasekaran","year":"2010","unstructured":"Shiv Chandrasekaran, Patrick Dewilde, Ming Gu, and Naveen Somasunderam, On the numerical rank of the off-diagonal blocks of Schur complements of discretized elliptic PDEs, SIAM J. Matrix Anal. Appl. 31 (2010), no.\u00a05, 2261\u20132290.","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"5","key":"9711_CR37","doi-asserted-by":"publisher","first-page":"A2974","DOI":"10.1137\/18M1207430","volume":"42","author":"Ke Chen","year":"2020","unstructured":"Ke\u00a0Chen, Qin Li, Jianfeng Lu, and Stephen\u00a0J Wright, Random sampling and efficient algorithms for multiscale PDEs, SIAM J. Sci. Comput. 42 (2020), no.\u00a05, A2974\u2013A3005.","journal-title":"SIAM J. Sci. Comput."},{"issue":"2","key":"9711_CR38","doi-asserted-by":"publisher","first-page":"980","DOI":"10.1137\/20M1352922","volume":"19","author":"Yifan Chen","year":"2021","unstructured":"Yifan Chen, Thomas\u00a0Y Hou, and Yixuan Wang, Exponential convergence for multiscale linear elliptic PDEs via adaptive edge basis functions, Multiscale Model. Simul. 19 (2021), no.\u00a02, 980\u20131010.","journal-title":"Multiscale Model. Simul."},{"key":"9711_CR39","doi-asserted-by":"crossref","unstructured":"Yifan Chen, Thomas\u00a0Y Hou, and Yixuan Wang, Exponentially convergent multiscale finite element method, Commun. Appl. Math. Comput. (2023), 1\u201317.","DOI":"10.1007\/s42967-023-00260-2"},{"issue":"3","key":"9711_CR40","doi-asserted-by":"publisher","first-page":"849","DOI":"10.1137\/22M1507802","volume":"21","author":"Yifan Chen","year":"2023","unstructured":"Yifan Chen, Thomas\u00a0Y Hou, and Yixuan Wang, Exponentially convergent multiscale methods for 2D high frequency heterogeneous Helmholtz equations, Multiscale Model. Simul. 21 (2023), no.\u00a03, 849\u2013883.","journal-title":"Multiscale Model. Simul."},{"key":"9711_CR41","doi-asserted-by":"crossref","unstructured":"Eric Chung, Yalchin Efendiev, and Thomas\u00a0Y Hou, Multiscale model reduction: multiscale finite element methods and their generalizations, vol. 212, Springer, 2023.","DOI":"10.1007\/978-3-031-20409-8"},{"key":"9711_CR42","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/s13137-014-0066-0","volume":"5","author":"Eric T Chung","year":"2014","unstructured":"Eric\u00a0T Chung, Yalchin Efendiev, and Shubin Fu, Generalized multiscale finite element method for elasticity equations, GEM Int. J. Geomath. 5 (2014), 225\u2013254.","journal-title":"GEM Int. J. Geomath."},{"key":"9711_CR43","doi-asserted-by":"publisher","first-page":"298","DOI":"10.1016\/j.cma.2018.04.010","volume":"339","author":"Eric T Chung","year":"2018","unstructured":"Eric\u00a0T Chung, Yalchin Efendiev, and Wing\u00a0Tat Leung, Constraint energy minimizing generalized multiscale finite element method, Comput. Methods Appl. Mech. Eng. 339 (2018), 298\u2013319.","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9711_CR44","doi-asserted-by":"publisher","first-page":"298","DOI":"10.1016\/j.cam.2018.08.054","volume":"348","author":"Eric T Chung","year":"2019","unstructured":"Eric\u00a0T Chung and Chak Shing Lee, A mixed generalized multiscale finite element method for planar linear elasticity, J. Comput. Appl. Math. 348 (2019), 298\u2013313.","journal-title":"J. Comput. Appl. Math."},{"issue":"273","key":"9711_CR45","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1090\/S0025-5718-2010-02353-1","volume":"80","author":"Alan Demlow","year":"2011","unstructured":"Alan Demlow, Johnny Guzm\u00e1n, and Alfred Schatz, Local energy estimates for the finite element method on sharply varying grids, Math. Comp. 80 (2011), no.\u00a0273, 1\u20139.","journal-title":"Math. Comp."},{"key":"9711_CR46","doi-asserted-by":"publisher","first-page":"116","DOI":"10.1016\/j.jcp.2013.04.045","volume":"251","author":"Yalchin Efendiev","year":"2013","unstructured":"Yalchin Efendiev, Juan Galvis, and Thomas\u00a0Y Hou, Generalized multiscale finite element methods (GMsFEM), J. Comput. Phys. 251 (2013), 116\u2013135.","journal-title":"J. Comput. Phys."},{"key":"9711_CR47","unstructured":"Yalchin Efendiev and Thomas\u00a0Y Hou, Multiscale finite element methods: theory and applications, vol.\u00a04, Springer Science & Business Media, 2009."},{"key":"9711_CR48","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1017\/S0962492906360011","volume":"17","author":"Bj\u00f6rn Engquist","year":"2008","unstructured":"Bj\u00f6rn Engquist and Panagiotis E Souganidis, Asymptotic and numerical homogenization, Acta Numerica 17 (2008), 147\u2013190.","journal-title":"Acta Numerica"},{"issue":"4","key":"9711_CR49","doi-asserted-by":"publisher","first-page":"1710","DOI":"10.1137\/07068583X","volume":"29","author":"Bj\u00f6rn Engquist","year":"2007","unstructured":"Bj\u00f6rn Engquist and Lexing Ying, Fast directional multilevel algorithms for oscillatory kernels, SIAM J. Sci. Comput. 29 (2007), no.\u00a04, 1710\u20131737.","journal-title":"SIAM J. Sci. Comput."},{"issue":"5","key":"9711_CR50","doi-asserted-by":"publisher","first-page":"697","DOI":"10.1002\/cpa.20358","volume":"64","author":"Bj\u00f6rn Engquist","year":"2011","unstructured":"Bj\u00f6rn Engquist and Lexing Ying, Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation, Commun. Pure Appl. Math. 64 (2011), no.\u00a05, 697\u2013735.","journal-title":"Commun. Pure Appl. Math."},{"issue":"11","key":"9711_CR51","doi-asserted-by":"publisher","first-page":"2220","DOI":"10.1002\/cpa.21755","volume":"71","author":"Bj\u00f6rn Engquist","year":"2018","unstructured":"Bj\u00f6rn Engquist and Hongkai Zhao, Approximate separability of the Green\u2019s function of the helmholtz equation in the high frequency limit, Commun. Pure Appl. Math. 71 (2018), no.\u00a011, 2220\u20132274.","journal-title":"Commun. Pure Appl. Math."},{"key":"9711_CR52","doi-asserted-by":"crossref","unstructured":"Markus Faustmann, Jens\u00a0Markus Melenk, and Maryam Parvizi, $${\\cal{H}}$$-matrix approximability of inverses of FEM matrices for the time-harmonic maxwell equations, Adv. Comput. Math. 48 (2022), no.\u00a05, 59.","DOI":"10.1007\/s10444-022-09965-z"},{"key":"9711_CR53","doi-asserted-by":"crossref","unstructured":"Markus Faustmann, Jens\u00a0Markus Melenk, and Dirk Praetorius, $$\\cal{H}$$-matrix approximability of the inverses of FEM matrices, Numer. Math. 131 (2015), no.\u00a04, 615\u2013642.","DOI":"10.1007\/s00211-015-0706-9"},{"issue":"1","key":"9711_CR54","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1007\/s00211-023-01386-4","volume":"156","author":"Philip Freese","year":"2024","unstructured":"Philip Freese, Moritz Hauck, Tim Keil, and Daniel Peterseim, A super-localized generalized finite element method, Numer. Math. 156 (2024), no.\u00a01, 205\u2013235.","journal-title":"Numer. Math."},{"issue":"4\u20136","key":"9711_CR55","doi-asserted-by":"publisher","first-page":"273","DOI":"10.1007\/s00791-008-0098-9","volume":"11","author":"Lars Grasedyck","year":"2008","unstructured":"Lars Grasedyck, Ronald Kriemann, and Sabine Le\u00a0Borne, Parallel black box-LU preconditioning for elliptic boundary value problems, Comput. Vis. Sci. 11 (2008), no.\u00a04-6, 273\u2013291.","journal-title":"Comput. Vis. Sci."},{"issue":"4","key":"9711_CR56","doi-asserted-by":"publisher","first-page":"565","DOI":"10.1007\/s00211-009-0218-6","volume":"112","author":"Lars Grasedyck","year":"2009","unstructured":"Lars Grasedyck, Ronald Kriemann, and Sabine Le\u00a0Borne, Domain decomposition based-LU preconditioning, Numer. Math. 112 (2009), no.\u00a04, 565\u2013600.","journal-title":"Numer. Math."},{"issue":"2","key":"9711_CR57","doi-asserted-by":"publisher","first-page":"325","DOI":"10.1016\/0021-9991(87)90140-9","volume":"73","author":"Leslie Greengard","year":"1987","unstructured":"Leslie Greengard and Vladimir Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys. 73 (1987), no.\u00a02, 325\u2013348.","journal-title":"J. Comput. Phys."},{"key":"9711_CR58","doi-asserted-by":"crossref","unstructured":"Wolfgang Hackbusch et\u00a0al., Hierarchical matrices: algorithms and analysis, vol.\u00a049, Springer, 2015.","DOI":"10.1007\/978-3-662-47324-5"},{"issue":"2","key":"9711_CR59","doi-asserted-by":"publisher","first-page":"311","DOI":"10.1051\/m2an\/2015046","volume":"50","author":"Christopher Harder","year":"2016","unstructured":"Christopher Harder, Alexandre\u00a0L Madureira, and Fr\u00e9d\u00e9ric Valentin, A hybrid-mixed method for elasticity, ESAIM: Math. Model. Numer. Anal. 50 (2016), no.\u00a02, 311\u2013336.","journal-title":"ESAIM: Math. Model. Numer. Anal."},{"issue":"341","key":"9711_CR60","doi-asserted-by":"publisher","first-page":"981","DOI":"10.1090\/mcom\/3798","volume":"92","author":"Moritz Hauck","year":"2023","unstructured":"Moritz Hauck and Daniel Peterseim, Super-localization of elliptic multiscale problems, Math. Comp. 92 (2023), no.\u00a0341, 981\u20131003.","journal-title":"Math. Comp."},{"key":"9711_CR61","doi-asserted-by":"publisher","first-page":"156","DOI":"10.1016\/j.cma.2016.06.034","volume":"310","author":"Patrick Henning","year":"2016","unstructured":"Patrick Henning and Anna Persson, A multiscale method for linear elasticity reducing poisson locking, Comput. Methods Appl. Mech. Eng. 310 (2016), 156\u2013171.","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"8","key":"9711_CR62","doi-asserted-by":"publisher","first-page":"1415","DOI":"10.1002\/cpa.21582","volume":"69","author":"Kenneth L Ho","year":"2016","unstructured":"Kenneth L Ho and Lexing Ying, Hierarchical interpolative factorization for elliptic operators: differential equations, Commun. Pure Appl. Math. 69 (2016), no.\u00a08, 1415\u20131451.","journal-title":"Commun. Pure Appl. Math."},{"issue":"227","key":"9711_CR63","doi-asserted-by":"publisher","first-page":"913","DOI":"10.1090\/S0025-5718-99-01077-7","volume":"68","author":"Thomas Hou","year":"1999","unstructured":"Thomas Hou, Xiao-Hui Wu, and Zhiqiang Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp. 68 (1999), no.\u00a0227, 913\u2013943.","journal-title":"Math. Comp."},{"issue":"1","key":"9711_CR64","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1006\/jcph.1997.5682","volume":"134","author":"Thomas Y Hou","year":"1997","unstructured":"Thomas Y Hou and Xiao-Hui Wu, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997), no.\u00a01, 169\u2013189.","journal-title":"J. Comput. Phys."},{"issue":"1\u20134","key":"9711_CR65","doi-asserted-by":"publisher","first-page":"387","DOI":"10.1016\/0045-7825(95)00844-9","volume":"127","author":"Thomas JR Hughes","year":"1995","unstructured":"Thomas\u00a0JR Hughes, Multiscale phenomena: Green\u2019s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Eng. 127 (1995), no.\u00a01-4, 387\u2013401.","journal-title":"Comput. Methods Appl. Mech. Eng."},{"issue":"2","key":"9711_CR66","doi-asserted-by":"publisher","first-page":"539","DOI":"10.1137\/050645646","volume":"45","author":"Thomas JR Hughes","year":"2007","unstructured":"Thomas\u00a0JR Hughes and Giancarlo Sangalli, Variational multiscale analysis: the fine-scale Green\u2019s function, projection, optimization, localization, and stabilized methods, SIAM J. Numer. Anal. 45 (2007), no.\u00a02, 539\u2013557.","journal-title":"SIAM J. Numer. Anal."},{"key":"9711_CR67","unstructured":"Vasili\u00a0Vasilievitch Jikov, Sergei\u00a0M Kozlov, and Olga\u00a0Arsenievna Oleinik, Homogenization of Differential Operators and Integral Functionals, Springer Science & Business Media, 2012."},{"issue":"3","key":"9711_CR68","doi-asserted-by":"publisher","first-page":"851","DOI":"10.1051\/m2an\/2016057","volume":"51","author":"Claude Le Bris","year":"2017","unstructured":"Claude Le\u00a0Bris, Frederic Legoll, and Fran\u00e7ois Madiot, A numerical comparison of some multiscale finite element approaches for advection-dominated problems in heterogeneous media, ESAIM: Math. Model. Numer. Anal. 51 (2017), no.\u00a03, 851\u2013888.","journal-title":"ESAIM: Math. Model. Numer. Anal."},{"key":"9711_CR69","doi-asserted-by":"crossref","unstructured":"Giovanni Leoni, A first course in sobolev spaces, vol. 181, American Mathematical Society, 2024.","DOI":"10.1090\/gsm\/229"},{"key":"9711_CR70","doi-asserted-by":"publisher","first-page":"378","DOI":"10.1016\/j.jcp.2014.03.066","volume":"270","author":"Songting Luo","year":"2014","unstructured":"Songting Luo, Jianliang Qian, and Robert Burridge, Fast huygens sweeping methods for Helmholtz equations in inhomogeneous media in the high frequency regime, J. Comput. Phys. 270 (2014), 378\u2013401.","journal-title":"J. Comput. Phys."},{"issue":"3","key":"9711_CR71","doi-asserted-by":"publisher","first-page":"1546","DOI":"10.1137\/21M1466748","volume":"61","author":"Chupeng Ma","year":"2023","unstructured":"Chupeng Ma, Christian Alber, and Robert Scheichl, Wavenumber explicit convergence of a multiscale generalized finite element method for heterogeneous Helmholtz problems, SIAM J. Numer. Anal. 61 (2023), no.\u00a03, 1546\u20131584.","journal-title":"SIAM J. Numer. Anal."},{"key":"9711_CR72","doi-asserted-by":"crossref","unstructured":"Chupeng Ma and Jens\u00a0Markus Melenk, Exponential convergence of a generalized FEM for heterogeneous reaction-diffusion equations, Multiscale Model. Simul. 22 (2024), no.\u00a01, 256\u2013282.","DOI":"10.1137\/22M1522231"},{"issue":"338","key":"9711_CR73","doi-asserted-by":"publisher","first-page":"2539","DOI":"10.1090\/mcom\/3755","volume":"91","author":"Chupeng Ma","year":"2022","unstructured":"Chupeng Ma and Robert Scheichl, Error estimates for discrete generalized FEMs with locally optimal spectral approximations, Math. Comp. 91 (2022), no.\u00a0338, 2539\u20132569.","journal-title":"Math. Comp."},{"issue":"1","key":"9711_CR74","doi-asserted-by":"publisher","first-page":"244","DOI":"10.1137\/21M1406179","volume":"60","author":"Chupeng Ma","year":"2022","unstructured":"Chupeng Ma, Robert Scheichl, and Tim Dodwell, Novel design and analysis of generalized finite element methods based on locally optimal spectral approximations, SIAM J. Numer. Anal. 60 (2022), no.\u00a01, 244\u2013273.","journal-title":"SIAM J. Numer. Anal."},{"issue":"3","key":"9711_CR75","doi-asserted-by":"publisher","first-page":"1064","DOI":"10.1137\/090775592","volume":"9","author":"Axel M\u00e5lqvist","year":"2011","unstructured":"Axel M\u00e5lqvist, Multiscale methods for elliptic problems, Multiscale Model. Simul. 9 (2011), no.\u00a03, 1064\u20131086.","journal-title":"Multiscale Model. Simul."},{"issue":"290","key":"9711_CR76","doi-asserted-by":"publisher","first-page":"2583","DOI":"10.1090\/S0025-5718-2014-02868-8","volume":"83","author":"Axel M\u00e5lqvist","year":"2014","unstructured":"Axel M\u00e5lqvist and Daniel Peterseim, Localization of elliptic multiscale problems, Math. Comp. 83 (2014), no.\u00a0290, 2583\u20132603.","journal-title":"Math. Comp."},{"key":"9711_CR77","doi-asserted-by":"crossref","unstructured":"Axel M\u00e5lqvist and Daniel Peterseim, Numerical homogenization by localized orthogonal decomposition, SIAM, 2020.","DOI":"10.1137\/1.9781611976458"},{"key":"9711_CR78","unstructured":"Jens\u00a0Markus Melenk, On generalized finite element methods, Ph.D. thesis, Department of Mathematics, University of Maryland, 1995."},{"key":"9711_CR79","first-page":"33","volume":"97","author":"ThV Nguyen","year":"1973","unstructured":"Th\u00a0V Nguyen, Bases communes pour certains espaces de fonctions harmoniques, Bull. Sc. Math 97 (1973), 33\u201349.","journal-title":"Bull. Sc. Math"},{"key":"9711_CR80","unstructured":"OA\u00a0Oleinik, AS\u00a0Shamaev, and GA\u00a0Yosifian, Mathematical problems in elasticity and homogenization, Elsevier, 1992."},{"issue":"3","key":"9711_CR81","doi-asserted-by":"publisher","first-page":"812","DOI":"10.1137\/140974596","volume":"13","author":"Houman Owhadi","year":"2015","unstructured":"Houman Owhadi, Bayesian numerical homogenization, Multiscale Model. Simul. 13 (2015), no.\u00a03, 812\u2013828.","journal-title":"Multiscale Model. Simul."},{"key":"9711_CR82","doi-asserted-by":"crossref","unstructured":"Houman Owhadi and Clint Scovel, Operator-adapted wavelets, fast solvers, and numerical homogenization: From a game theoretic approach to numerical approximation and algorithm design, vol.\u00a035, Cambridge University Press, 2019.","DOI":"10.1017\/9781108594967"},{"issue":"4","key":"9711_CR83","doi-asserted-by":"publisher","first-page":"1373","DOI":"10.1137\/100813968","volume":"9","author":"Houman Owhadi","year":"2011","unstructured":"Houman Owhadi and Lei Zhang, Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast, Multiscale Model. Simul. 9 (2011), no.\u00a04, 1373\u20131398.","journal-title":"Multiscale Model. Simul."},{"key":"9711_CR84","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1016\/j.jcp.2017.06.037","volume":"347","author":"Houman Owhadi","year":"2017","unstructured":"Houman Owhadi and Lei Zhang, Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs\/PDEs with rough coefficients, J. Comput. Phys. 347 (2017), 99\u2013128.","journal-title":"J. Comput. Phys."},{"issue":"01","key":"9711_CR85","doi-asserted-by":"publisher","first-page":"17","DOI":"10.1142\/S0219876204000071","volume":"1","author":"Peter J Park","year":"2004","unstructured":"Peter J Park and Thomas Y Hou, Multiscale numerical methods for singularly perturbed convection-diffusion equations, Int. J. Comput. Methods 1 (2004), no.\u00a001, 17\u201365.","journal-title":"Int. J. Comput. Methods"},{"issue":"1","key":"9711_CR86","doi-asserted-by":"publisher","first-page":"89","DOI":"10.1007\/BF00277432","volume":"8","author":"LE Payne","year":"1961","unstructured":"LE\u00a0Payne and Hans F Weinberger, On Korn\u2019s inequality, Arch. Ration. Mech. Anal. 8 (1961), no.\u00a01, 89\u201398.","journal-title":"Arch. Ration. Mech. Anal."},{"key":"9711_CR87","doi-asserted-by":"crossref","unstructured":"Allan Pinkus, n-widths in approximation theory, Springer-Verlag, Berlin, 1985.","DOI":"10.1007\/978-3-642-69894-1"},{"issue":"1","key":"9711_CR88","doi-asserted-by":"publisher","first-page":"551","DOI":"10.1137\/20M1384294","volume":"20","author":"Julia Schleu\u00df","year":"2022","unstructured":"Julia Schleu\u00df and Kathrin Smetana, Optimal local approximation spaces for parabolic problems, Multiscale Model. Simul. 20 (2022), no.\u00a01, 551\u2013582.","journal-title":"Multiscale Model. Simul."},{"issue":"2","key":"9711_CR89","doi-asserted-by":"publisher","first-page":"175","DOI":"10.1006\/jath.1999.3402","volume":"102","author":"N Skiba","year":"2000","unstructured":"N\u00a0Skiba and V\u00a0Zahariuta, Asymptotics of Kolmogorov diameters for some classes of harmonic functions on spheroids, J. Approx. Theory 102 (2000), no.\u00a02, 175\u2013188.","journal-title":"J. Approx. Theory"},{"issue":"5","key":"9711_CR90","doi-asserted-by":"publisher","first-page":"A3318","DOI":"10.1137\/15M1009603","volume":"38","author":"Kathrin Smetana","year":"2016","unstructured":"Kathrin Smetana and Anthony T Patera, Optimal local approximation spaces for component-based static condensation procedures, SIAM J. Sci. Comput. 38 (2016), no.\u00a05, A3318\u2013A3356.","journal-title":"SIAM J. Sci. Comput."},{"issue":"32\u201333","key":"9711_CR91","doi-asserted-by":"publisher","first-page":"4081","DOI":"10.1016\/S0045-7825(01)00188-8","volume":"190","author":"Theofanis Strouboulis","year":"2001","unstructured":"Theofanis Strouboulis, Kevin Copps, and Ivo Babu\u0161ka, The generalized finite element method, Comput. Methods Appl. Mech. Eng. 190 (2001), no.\u00a032-33, 4081\u20134193.","journal-title":"Comput. Methods Appl. Mech. Eng."},{"key":"9711_CR92","doi-asserted-by":"crossref","unstructured":"Andrea Toselli and Olof Widlund, Domain decomposition methods-algorithms and theory, vol.\u00a034, Springer Science & Business Media, 2004.","DOI":"10.1007\/b137868"},{"issue":"1","key":"9711_CR93","doi-asserted-by":"publisher","first-page":"87","DOI":"10.4310\/CMS.2003.v1.n1.a8","volume":"1","author":"E Weinan","year":"2003","unstructured":"E\u00a0Weinan and Bjorn Engquist, The heterognous multiscale methods, Commun. Math. Sci. 1 (2003), no.\u00a01, 87\u2013132.","journal-title":"Commun. Math. Sci."},{"issue":"3","key":"9711_CR94","doi-asserted-by":"publisher","first-page":"1277","DOI":"10.1093\/imanum\/dru033","volume":"35","author":"Bingxing Xia","year":"2015","unstructured":"Bingxing Xia and Viet Ha Hoang, High-dimensional finite element method for multiscale linear elasticity, IMA J. Numer. Anal. 35 (2015), no.\u00a03, 1277\u20131314.","journal-title":"IMA J. Numer. Anal."},{"issue":"2","key":"9711_CR95","doi-asserted-by":"publisher","first-page":"A832","DOI":"10.1137\/120867032","volume":"35","author":"Jianlin Xia","year":"2013","unstructured":"Jianlin Xia, Efficient structured multifrontal factorization for general large sparse matrices, SIAM J. Sci. Comput. 35 (2013), no.\u00a02, A832\u2013A860.","journal-title":"SIAM J. Sci. Comput."},{"issue":"6","key":"9711_CR96","doi-asserted-by":"publisher","first-page":"953","DOI":"10.1002\/nla.691","volume":"17","author":"Jianlin Xia","year":"2010","unstructured":"Jianlin Xia, Shivkumar Chandrasekaran, Ming Gu, and Xiaoye S Li, Fast algorithms for hierarchically semiseparable matrices, Numer. Linear Algebra Appl. 17 (2010), no.\u00a06, 953\u2013976.","journal-title":"Numer. Linear Algebra Appl."},{"issue":"3","key":"9711_CR97","doi-asserted-by":"publisher","first-page":"1382","DOI":"10.1137\/09074543X","volume":"31","author":"Jianlin Xia","year":"2010","unstructured":"Jianlin Xia, Shivkumar Chandrasekaran, Ming Gu, and Xiaoye S Li, Superfast multifrontal method for large structured linear systems of equations, SIAM J. Matrix Anal. Appl. 31 (2010), no.\u00a03, 1382\u20131411.","journal-title":"SIAM J. Matrix Anal. Appl."},{"issue":"2","key":"9711_CR98","doi-asserted-by":"publisher","first-page":"735","DOI":"10.1137\/22M1487655","volume":"21","author":"Lina Zhao","year":"2023","unstructured":"Lina Zhao and Eric Chung, Constraint energy minimizing generalized multiscale finite element method for convection diffusion equation, Multiscale Model. Simul. 21 (2023), no.\u00a02, 735\u2013752.","journal-title":"Multiscale Model. Simul."}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-025-09711-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-025-09711-z","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-025-09711-z.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,6,1]],"date-time":"2026-06-01T17:01:45Z","timestamp":1780333305000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-025-09711-z"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,4,29]]},"references-count":98,"journal-issue":{"issue":"3","published-print":{"date-parts":[[2026,6]]}},"alternative-id":["9711"],"URL":"https:\/\/doi.org\/10.1007\/s10208-025-09711-z","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,4,29]]},"assertion":[{"value":"11 March 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 March 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"25 March 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"29 April 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}