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Madureira, Alexandre L.; Sarkis, Marcus (2017)
Languages: English
Types: Preprint
Subjects: Mathematics - Numerical Analysis
We consider finite element methods of multiscale type to approximate solutions for two-dimensional symmetric elliptic partial differential equations with heterogeneous $L^\infty$ coefficients. The methods are of Galerkin type and follows the Variational Multiscale and Localized Orthogonal Decomposition--LOD approaches in the sense that it decouples spaces into multiscale and fine subspaces. In a first method, the multiscale basis functions are obtained by mapping coarse basis functions, based on corners used on primal iterative substructuring methods, to functions of global minimal energy. This approach delivers quasi-optimal a priori error energy approximation with respect to the mesh size, however it deteriorates with respect to high-contrast coefficients. In a second method, edge modes based on local generalized eigenvalue problems are added to the corner modes. As a result, optimal a priori error energy estimate is achieved which is mesh and contrast independent. The methods converge at optimal rate even if the solution has minimum regularity, belonging only to the Sobolev space $H^1$.
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    • [1] Rodolfo Araya, Christopher Harder, Diego Paredes, and Fr´ed´eric Valentin, Multiscale hybrid-mixed method, SIAM J. Numer. Anal. 51 (2013), no. 6, 3505-3531, DOI 10.1137/120888223. MR3143841
    • [2] Ivo Babuˇska, Gabriel Caloz, and John E. Osborn, Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31 (1994), no. 4, 945-981. MR1286212 (95g:65146)
    • [3] Ivo Babuska and Robert Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul. 9 (2011), no. 1, 373-406, DOI 10.1137/100791051. MR2801210
    • [4] L. Beira˜o da Veiga, L. F. Pavarino, S. Scacchi, O. B. Widlund, and S. Zampini, Adaptive selection of primal constraints for isogeometric BDDC deluxe preconditioners, SIAM J. Sci. Comput. 39 (2017), no. 1, A281-A302, DOI 10.1137/15M1054675. MR3612901
    • [5] Petter E. Bjørstad and Olof B. Widlund, Solving elliptic problems on regions partitioned into substructures, Elliptic problem solvers, II (Monterey, Calif., 1983), Academic Press, Orlando, FL, 1984, pp. 245-255. MR764237
    • [6] F. Bourquin, Component mode synthesis and eigenvalues of second order operators: discretization and algorithm, RAIRO Mod´el. Math. Anal. Num´er. 26 (1992), no. 3, 385-423, DOI 10.1051/m2an/1992260303851 (English, with French summary). MR1160133
    • [7] Fr´ed´eric Bourquin, Analysis and comparison of several component mode synthesis methods on onedimensional domains, Numer. Math. 58 (1990), no. 1, 11-33, DOI 10.1007/BF01385608. MR1069651
    • [8] J. H. Bramble, J. E. Pasciak, and A. H. Schatz, The construction of preconditioners for elliptic problems by substructuring. I, Math. Comp. 47 (1986), no. 175, 103-134, DOI 10.2307/2008084. MR842125
    • [9] Juan G. Calvo and Olof B. Widlund, An adaptive choice of primal constraints for BDDC domain decomposition algorithms, Electron. Trans. Numer. Anal. 45 (2016), 524-544. MR3582898
    • [10] C.-C. Chu, I. G. Graham, and T.-Y. Hou, A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp. 79 (2010), no. 272, 1915-1955, DOI 10.1090/S0025-5718-2010- 02372-5. MR2684351
    • [11] Eric T Chung, Yalchin Efendiev, and Wing Tat Leung, Constraint energy minimizing generalized multiscale finite element method, ArXiv e-prints (2017), available at 1704.03193.
    • [12] , Constraint energy minimizing generalized multiscale finite element method in the mixed formulation, ArXiv e-prints (2017), available at 1705.05959.
    • [13] Roy R Craig and Mervyn CC Bampton, Coupling of substructures for dynamic analysis, AIAA journal 6 (1968), no. 7, 1313-1319.
    • [14] Clark R. Dohrmann, A preconditioner for substructuring based on constrained energy minimization, SIAM J. Sci. Comput. 25 (2003), no. 1, 246-258, DOI 10.1137/S1064827502412887. MR2047204
    • [15] C. Dohrmann and C. Pechstein, Modern decomposition solvers - BDDC, deluxe scaling, and an algebraic approach, slides to a talk at NuMa Seminar, JKU Linz, Linz, Austria, December 10, 2013, http://people.ricam.oeaw.ac.at/c.pechstein/pechstein-bddc2013.pdf.
    • [16] , A unified framework for adaptive BDDC, Technical Report 2016-20, Johann Radon Institute for Computational and Applied Mathematics (RICAM) (2016).
    • [17] Victorita Dolean, Fr´ed´eric Nataf, Robert Scheichl, and Nicole Spillane, Analysis of a two-level Schwarz method with coarse spaces based on local Dirichlet-to-Neumann maps, Comput. Methods Appl. Math. 12 (2012), no. 4, 391-414, DOI 10.2478/cmam-2012-0027. MR3033238
    • [18] Zdenˇek Dost´al, David Hora´k, and Radek Kuˇcera, Total FETI-an easier implementable variant of the FETI method for numerical solution of elliptic PDE, Comm. Numer. Methods Engrg. 22 (2006), no. 12, 1155-1162, DOI 10.1002/cnm.881. MR2282408 (2007k:65177)
    • [19] Maksymilian Dryja, Marcus V. Sarkis, and Olof B. Widlund, Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math. 72 (1996), no. 3, 313-348, DOI 10.1007/s002110050172. MR1367653
    • [20] Maksymilian Dryja, Barry F. Smith, and Olof B. Widlund, Schwarz analysis of iterative substructuring algorithms for elliptic problems in three dimensions, SIAM J. Numer. Anal. 31 (1994), no. 6, 1662-1694, DOI 10.1137/0731086. MR1302680
    • [21] Maksymilian Dryja and Marcus Sarkis, Technical tools for boundary layers and applications to heterogeneous coefficients, Domain decomposition methods in science and engineering XIX, Lect. Notes Comput. Sci. Eng., vol. 78, Springer, Heidelberg, 2011, pp. 205-212, DOI 10.1007/978-3-642-11304- 8 22. MR2867661
    • [22] Weinan E and Bjorn Engquist, The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87-132.MR1979846 (2004b:35019)
    • [23] Weinan E and Pingbing Ming, Analysis of multiscale methods, J. Comput. Math. 22 (2004), no. 2, 210-219.MR2058933 (2005d:65188)
    • [24] Y. Efendiev, J. Galvis, R. Lazarov, M. Moon, and M. Sarkis, Generalized multiscale finite element method. Symmetric interior penalty coupling, J. Comput. Phys. 255 (2013), 1-15, DOI 10.1016/j.jcp.2013.07.028. MR3109775
    • [25] Yalchin Efendiev, Juan Galvis, and Xiao-Hui Wu, Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comput. Phys. 230 (2011), no. 4, 937-955, DOI 10.1016/j.jcp.2010.09.026. MR2753343
    • [26] Yalchin Efendiev, Juan Galvis, Raytcho Lazarov, and Joerg Willems, Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms, ESAIM Math. Model. Numer. Anal. 46 (2012), no. 5, 1175-1199, DOI 10.1051/m2an/2011073. MR2916377
    • [27] Charbel Farhat and Francois-Xavier Roux, A method of finite element tearing and interconnecting and its parallel solution algorithm, Int. J. Numer. Meth. Engng. 32 (1991), no. 6, 1205-1227, DOI 10.1002/nme.1620320604.
    • [28] Charbel Farhat, Michael Lesoinne, and Kendall Pierson, A scalable dual-primal domain decomposition method, Numer. Linear Algebra Appl. 7 (2000), no. 7-8, 687-714, DOI 10.1002/1099- 1506(200010/12)7:7/8¡687::AID-NLA219¿3.0.CO;2-S. Preconditioning techniques for large sparse matrix problems in industrial applications (Minneapolis, MN, 1999). MR1802366
    • [29] Charbel Farhat, Jan Mandel, and Franc¸ois-Xavier Roux, Optimal convergence properties of the FETI domain decomposition method, Comput. Methods Appl. Mech. Engrg. 115 (1994), no. 3-4, 365-385, DOI 10.1016/0045-7825(94)90068-X. MR1285024
    • [30] Juan Galvis and Yalchin Efendiev, Domain decomposition preconditioners for multiscale flows in high-contrast media, Multiscale Model. Simul. 8 (2010), no. 4, 1461-1483, DOI 10.1137/090751190. MR2718268
    • [31] Johnny Guzma´n, Manuel A. Sa´nchez, and Marcus Sarkis, A finite element method for high-contrast interface problems with error estimates independent of contrast (2016), accepted for Journal of Scientific Computing.
    • [32] Christopher Harder, Alexandre Madureira, and Fr´ed´eric Valentin, A Hybrid-Mixed Method for Elasticity 2 in Two and Three-Dimensions (2015), accepted for publication at M AN.
    • [33] Christopher Harder, Diego Paredes, and Fr´ed´eric Valentin, A family of multiscale hybrid-mixed finite element methods for the Darcy equation with rough coefficients, J. Comput. Phys. 245 (2013), 107-130.
    • [34] Fredrik Hellman and Axel M˚alqvist, Contrast independent localization of multiscale problems, ArXiv e-prints (2016), available at 1610.07398.
    • [35] A. Heinlein, U. Hetmaniuk, A. Klawonn, and O. Rheinbach, The approximate component mode synthesis special finite element method in two dimensions: parallel implementation and numerical results, J. Comput. Appl. Math. 289 (2015), 116-133, DOI 10.1016/j.cam.2015.02.053. MR3350765
    • [36] Ulrich L. Hetmaniuk and Richard B. Lehoucq, A special finite element method based on component mode synthesis, M2AN Math. Model. Numer. Anal. 44 (2010), no. 3, 401-420, DOI 10.1051/m2an/2010007. MR2666649
    • [37] Ulrich Hetmaniuk and Axel Klawonn, Error estimates for a two-dimensional special finite element method based on component mode synthesis, Electron. Trans. Numer. Anal. 41 (2014), 109-132. MR3225627
    • [38] Thomas Y. 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. 227, 913- 943.MR1642758 (99i:65126)
    • [39] Thomas J. R. Hughes, Gonzalo R. Feij´oo, Luca Mazzei, and Jean-Baptiste Quincy, The variational multiscale method-a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998), no. 1-2, 3-24, DOI 10.1016/S0045-7825(98)00079-6. MR1660141 (99m:65239)
    • [40] T. J. R. Hughes and G. Sangalli, Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods, SIAM J. Numer. Anal. 45 (2007), no. 2, 539-557, DOI 10.1137/050645646. MR2300286 (2008c:65332)
    • [41] Walter C Hurty, Vibrations of structural systems by component mode synthesis, Journal of the Engineering Mechanics Division 86 (1960), no. 4, 51-70.
    • [42] Hyea Hyun Kim and Eric T. Chung, A BDDC algorithm with enriched coarse spaces for two-dimensional elliptic problems with oscillatory and high contrast coefficients, Multiscale Model. Simul. 13 (2015), no. 2, 571-593, DOI 10.1137/140970598. MR3350292
    • [43] A. Klawonn, P. Radtke, and O. Rheinbach, FETI-DP methods with an adaptive coarse space, SIAM J. Numer. Anal. 53 (2015), no. 1, 297-320, DOI 10.1137/130939675. MR3303686
    • [44] Axel Klawonn, Martin Ku¨hn, and Oliver Rheinbach, Adaptive coarse spaces for FETI-DP in three dimensions, SIAM J. Sci. Comput. 38 (2016), no. 5, A2880-A2911, DOI 10.1137/15M1049610. MR3546980
    • [45] Axel Klawonn, Olof B. Widlund, and Maksymilian Dryja, Dual-primal FETI methods for threedimensional elliptic problems with heterogeneous coefficients, SIAM J. Numer. Anal. 40 (2002), no. 1, 159-179, DOI 10.1137/S0036142901388081. MR1921914
    • [46] A. V. Knyazev and A. L. Skorokhodov, On exact estimates of the convergence rate of the steepest ascent method in the symmetric eigenvalue problem, Linear Algebra Appl. 154/156 (1991), 245-257, DOI 10.1016/0024-3795(91)90379-B. MR1113145
    • [47] A. V. Knyazev, A preconditioned conjugate gradient method for eigenvalue problems and its implementation in a subspace, Numerical treatment of eigenvalue problems, Vol. 5 (Oberwolfach, 1990), Internat. Ser. Numer. Math., vol. 96, Birkh¨auser, Basel, 1991, pp. 143-154. MR1109101
    • [48] T. Kozubek, V. Vondra´k, M. Menˇsık, D. Hora´k, Z. Dost´al, V. Hapla, P. Kabelıkov´a, and M. Cˇerm´ak, Total FETI domain decomposition method and its massively parallel implementation, Advances in Engineering Software 60-61 (2013), 14-22, DOI 10.1016/j.advengsoft.2013.04.001.
    • [49] A. L. Madureira and M. Sarkis, Hybrid Localized Spectral Decomposition for multiscale problems, ArXiv e-prints (2017), available at 1706.08941.
    • [50] Axel M˚alqvist and Daniel Peterseim, Localization of elliptic multiscale problems, Math. Comp. 83 (2014), no. 290, 2583-2603, DOI 10.1090/S0025-5718-2014-02868-8. MR3246801
    • [51] Axel M˚alqvist, Multiscale methods for elliptic problems, Multiscale Model. Simul. 9 (2011), no. 3, 1064- 1086, DOI 10.1137/090775592. MR2831590 (2012j:65419)
    • [52] Axel M˚alqvist, Patrick Henning, and Fredrik Hellman, Multiscale mixed finite elements, Discrete Contin. Dyn. Syst. Ser. S 9 (2016), no. 5, 1269-1298, DOI 10.3934/dcdss.2016051. MR3591945
    • [53] Jan Mandel and Clark R. Dohrmann, Convergence of a balancing domain decomposition by constraints and energy minimization, Numer. Linear Algebra Appl. 10 (2003), no. 7, 639-659, DOI 10.1002/nla.341. Dedicated to the 70th birthday of Ivo Marek. MR2030628
    • [54] Jan Mandel and Bedˇrich Soused´ık, Adaptive selection of face coarse degrees of freedom in the BDDC and the FETI-DP iterative substructuring methods, Comput. Methods Appl. Mech. Engrg. 196 (2007), no. 8, 1389-1399, DOI 10.1016/j.cma.2006.03.010. MR2277024
    • [55] , Adaptive coarse space selection in the BDDC and the FETI-DP iterative substructuring methods: optimal face degrees of freedom, Domain decomposition methods in science and engineering XVI, Lect. Notes Comput. Sci. Eng., vol. 55, Springer, Berlin, 2007, pp. 421-428, DOI 10.1007/978-3-540-34469- 8 52. MR2334131
    • [56] Alfio Quarteroni and Alberto Valli, Domain decomposition methods for partial differential equations, Numerical Mathematics and Scientific Computation, The Clarendon Press, Oxford University Press, New York, 1999. Oxford Science Publications. MR1857663
    • [57] D-S. Oh, O. B Widlund, S. Zampini, and C. R. Dohrmann, BDDC Algorithms with deluxe scaling and adaptive selection of primal constraints for Raviart-Thomas vector fields, Math. Comp. (2017), to appear.
    • [58] Houman Owhadi, Lei Zhang, and Leonid Berlyand, Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 2, 517-552, DOI 10.1051/m2an/2013118. MR3177856
    • [59] Clemens Pechstein, Finite and boundary element tearing and interconnecting solvers for multiscale problems, Lecture Notes in Computational Science and Engineering, vol. 90, Springer, Heidelberg, 2013. MR3013465
    • [60] Clemens Pechstein and Robert Scheichl, Weighted Poincar´e inequalities, IMA J. Numer. Anal. 33 (2013), no. 2, 652-686, DOI 10.1093/imanum/drs017. MR3047947
    • [61] , Analysis of FETI methods for multiscale PDEs, Numer. Math. 111 (2008), no. 2, 293-333, DOI 10.1007/s00211-008-0186-2. MR2456834
    • [62] , Analysis of FETI methods for multiscale PDEs. Part II: interface variation, Numer. Math. 118 (2011), no. 3, 485-529, DOI 10.1007/s00211-011-0359-2. MR2810804
    • [63] Daniel Peterseim and Robert Scheichl, Robust numerical upscaling of elliptic multiscale problems at high contrast, Comput. Methods Appl. Math. 16 (2016), no. 4, 579-603, DOI 10.1515/cmam-2016-0022. MR3552482
    • [64] P.-A. Raviart and J. M. Thomas, Primal hybrid finite element methods for 2nd order elliptic equations, Math. Comp. 31 (1977), no. 138, 391-413. MR0431752 (55 #4747)
    • [65] Giancarlo Sangalli, Capturing small scales in elliptic problems using a residual-free bubbles finite element method, Multiscale Model. Simul. 1 (2003), no. 3, 485-503 (electronic). MR2030161 (2004m:65202)
    • [66] Marcus Sarkis, Nonstandard coarse spaces and Schwarz methods for elliptic problems with discontinuous coefficients using non-conforming elements, Numer. Math. 77 (1997), no. 3, 383-406, DOI 10.1007/s002110050292. MR1469678
    • [67] Marcus Sarkis and Henrique Versieux, Convergence analysis for the numerical boundary corrector for elliptic equations with rapidly oscillating coefficients, SIAM J. Numer. Anal. 46 (2008), no. 2, 545-576. MR2383203
    • [68] Robert Scheichl, Panayot S. Vassilevski, and Ludmil T. Zikatanov, Weak approximation properties of elliptic projections with functional constraints, Multiscale Model. Simul. 9 (2011), no. 4, 1677-1699, DOI 10.1137/110821639. MR2861254
    • [69] N. Spillane, V. Dolean, P. Hauret, F. Nataf, C. Pechstein, and R. Scheichl, Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps, Numer. Math. 126 (2014), no. 4, 741-770, DOI 10.1007/s00211-013-0576-y. MR3175183
    • [70] Nicole Spillane, Victorita Dolean, Patrice Hauret, Fr´ed´eric Nataf, and Daniel J. Rixen, Solving generalized eigenvalue problems on the interfaces to build a robust two-level FETI method, C. R. Math. Acad. Sci. Paris 351 (2013), no. 5-6, 197-201, DOI 10.1016/j.crma.2013.03.010 (English, with English and French summaries). MR3089678
    • [71] Andrea Toselli and Olof Widlund, Domain decomposition methods-algorithms and theory, Springer Series in Computational Mathematics, vol. 34, Springer-Verlag, Berlin, 2005. MR2104179 (2005g:65006)
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