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Multiscale-spectral GFEM and optimal oversampling. (English) Zbl 1442.65402

Summary: In this work we address the Multiscale Spectral Generalized Finite Element Method (MS-GFEM) developed in [the first author and the second author, Multiscale Model. Simul. 9, No. 1, 373–406 (2011; Zbl 1229.65195)]. We outline the numerical implementation of this method and present simulations that demonstrate contrast independent exponential convergence of MS-GFEM solutions. We introduce strategies to reduce the computational cost of generating the optimal oversampled local approximating spaces used here. These strategies retain accuracy while reducing the computational work necessary to generate local bases. Motivated by oversampling we develop a nearly optimal local basis based on a partition of unity on the boundary and the associated A-harmonic extensions.

MSC:

65N35 Spectral, collocation and related methods for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs

Citations:

Zbl 1229.65195
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References:

[1] Babuška, I.; Caloz, G.; Osborn, J. E., Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal., 31, 945-981 (1994) · Zbl 0807.65114
[2] Babuška, I.; Banerjee, U.; Osborn, J. E., Survey of meshless and generalized finite element methods: a unified approach, Acta Numer., 12, 1-125 (2003) · Zbl 1048.65105
[3] Babuška, I.; Melenk, J., The partition of unity finite element method, Internat. J. Numer. Methods Engrg., 40, 727-758 (1997) · Zbl 0949.65117
[4] Strouboulis, T.; Babuška, I.; Copps, K., The design and analysis of the generalized finite element method, Comput. Methods Appl. Mech. Engrg., 181, 43-69 (2001) · Zbl 0983.65127
[5] Melenk, J. M.; Babuška, I., The partition of unity finite element method: basic theory and applications, Comput. Methods Appl. Mech. Engrg., 39, 289-314 (1996) · Zbl 0881.65099
[6] Babuška, I.; Lipton, R., Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul., 9, 373-406 (2011) · Zbl 1229.65195
[7] Babuška, I.; Huang, X.; Lipton, R., Machine computation using the exponentially convergent multiscale spectral generalized finite element method, ESAIM Math. Model. Numer. Anal., 48, 493-515 (2014) · Zbl 1320.74097
[8] Babuška, I., On the Schwarz algorithm in the theory of differential equations of Mathematical Physics, Tchecsl. Math. J., 8, 83, 328-342 (1958), (in Russian)
[9] Lions, P. L., On the Schwarz alternating method. I, (1st International Symposium on Domain Decomposition Methods for Partial Differential Equations (1988), SIAM: SIAM Philadelphia), 1-42 · Zbl 0658.65090
[10] Strouboulis, T.; Zhang, L.; Babuška, I., Generalized finite element method using mesh-based handbooks: application to problems in domains with many voids, Comput. Methods Appl. Mech. Engrg., 192, 3109-3161 (2003) · Zbl 1054.74059
[11] Schweitzer, M. A.; Wu, S., Evaluation of local multiscale approximation spaces for partition of unity methods (2017), preprint · Zbl 1446.74081
[12] Babuška, I.; Lipton, R.; Stuebner, M., The penetration function and its application to microscale problems, BIT Numer. Math., 48, 167-187 (2008) · Zbl 1152.65105
[13] Lipton, R.; Sinz, P.; Stuebner, M., Uncertain loading and quantifying maximum energy concentration within composite structures, J. Comput. Phys., 325, 38-52 (2016) · Zbl 1375.74038
[14] Halko, N.; Martinsson, P. G.; Tropp, J. A., Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53, 217-288 (2011) · Zbl 1269.65043
[15] K. Chen, Q. Li, J. Lu, S.J. Wright, Randomized sampling for basis functions construction in generalized finite element methods, arXiv:1801.06938, preprint. · Zbl 1446.65162
[16] Bhur, A.; Smetana, K., Randomized local model order reduction, SIAM Sci. Comput., 40, A2120-A2151 (2018) · Zbl 1394.65135
[17] Besounssan, A.; Lions, J. L.; Papanicolau, G. C., Asymptotic Analysis for Periodic Structures (1978), North Holland Pub.: North Holland Pub. Amsterdam · Zbl 0404.35001
[18] Hou, T. Y.; Wu, Xiao-Hui, A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys., 134, 169-189 (1997) · Zbl 0880.73065
[19] Hughes, T. J.R.; Feijoo, G. R.; Mazzei, L.; Quincy, J. B., The variational multiscale method. A Paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg., 166, 3-24 (1998) · Zbl 1017.65525
[20] Owhadi, H.; Zhang, L., Metric-based upscaling, Commun. Pure Appl. Math., 60, 675-723 (2007) · Zbl 1190.35070
[21] Owhadi, H.; Zhang, L., Homogenization of parabolic equations with a continuum of space and time scales, SIAM J. Numer. Anal., 46, 1-36 (2007) · Zbl 1170.34037
[22] Bebendorf, M., Why finite element discretizations can be factored by triangular hierarchical ma- trices, SIAM J. Numer. Anal., 45, 1472-1494 (2007) · Zbl 1152.65042
[23] Hackbusch, W., (Hierarchical Matrices: Algorithms and Analysis. Hierarchical Matrices: Algorithms and Analysis, Springer Series in Computational Mathematics (2015), Springer) · Zbl 1336.65041
[24] Owhadi, H., BayesIan numerical homogenization, Multiscale Model. Simul., 13, 812-828 (2015) · Zbl 1322.35002
[25] Efendiev, Y.; Ginting, V.; Hou, T.; Ewing, R., Accurate multiscale finite element methods for two-phase flow simulations, J. Comput. Phys., 220, 155-174 (2006) · Zbl 1158.76349
[26] Efendiev, Y.; Hou, T., Multiscale finite element methods for porous media flows and their applications, Appl. Numer. Math., 57, 577-596 (2007) · Zbl 1112.76046
[27] E, W.; Engquist, B.; Li, X.; Ren, W.; Vanden-Eijnden, E., Heterogeneous multiscale methods: A review, Commun. Comput. Phys., 2, 367-450 (2007) · Zbl 1164.65496
[28] E, W.; Ming, P.; Zhang, P., Analysis of the heterogeneous multiscale method for elliptic homogenization problems, J. Amer. Math. Soc., 18, 121-156 (2005) · Zbl 1060.65118
[29] Engquist, B.; Souganidis, P. E., Asymptotic and numerical homogenization, Acta Numer., 17, 147-190 (2008) · Zbl 1179.65142
[30] Nolen, J.; Papanicolaou, G.; Pironneau, O., A framework for adaptive multiscale methods for elliptic problems, Multiscale Model. Simul., 7, 171-196 (2008) · Zbl 1160.65342
[31] Berly, L.; Owhadi, H., Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Arch. Ration. Mech. Anal., 198, 677-721 (2010) · Zbl 1229.35009
[32] Owhadi, H.; Zhang, L.; Berlyand, L., Polyharmonic homogenization rough polyharmonic splines and sparse super-localization, ESAIM Math. Model. Numer. Anal., 48, 2, 517-552 (2014) · Zbl 1296.41007
[33] Arbogast, T.; Boyd, K. J., Subgrid upscaling and mixed multiscale finite elements, SIAM J. Numer. Anal., 44, 1150-1171 (2006) · Zbl 1120.65122
[34] Melenk, J. M., On n-widths for elliptic problems, J. Math. Anal. Appl., 247, 272-289 (2000) · Zbl 0963.35047
[35] Malquivst, A.; Peterseim, D., Localization of elliptic multiscale problems, Math. Comput., 83, 2583-2603 (2014) · Zbl 1301.65123
[36] Griebel, M.; Schweitzer, M. A., A particle-partition of unity method part VII: Adaptivity, (Meshfree Methods for Partial Differential Equations III, Vol. 57 (2007), Springer), 121-147 · Zbl 1119.65419
[37] Babuška, I.; Banerjee, U.; Osborn, J. E., Generalized finite element methods-main ideas, results and perspective, Int. J. Comput. Methods, 1, 67-103 (2004) · Zbl 1081.65107
[38] Dacorogna, B., Direct Methods in the Calculus of Variations (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0703.49001
[39] Pinkus, A., n-Widths in Approximation Theory (1985), Springer-Verlag: Springer-Verlag Berlin · Zbl 0551.41001
[40] Mathew, T. P.A., (Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations. Domain Decomposition Methods for the Numerical Solution of Partial Differential Equations, Lecture Notes in Computational Science and Engineering, vol. 61 (2008), Springer-Verlag: Springer-Verlag Berlin) · Zbl 1147.65101
[41] Toselli, A.; Widlund, O., (Domain Decomposition Methods-Algorithms and Theory. Domain Decomposition Methods-Algorithms and Theory, Springer Series in Computational Mathematics, vol. 34 (2005), Springer Verlag: Springer Verlag Berlin) · Zbl 1069.65138
[42] Dolean, V.; Jolivet, P.; Nataf, F., An Introduction to Domain Decomposition Methods: Algorithms, Theory and Parallel Implementation (2015), SIAM: SIAM Philadelphia · Zbl 1364.65277
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