Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-10T10:24:12.776Z Has data issue: false hasContentIssue false
  • English
  • Français

Numerical homogenization beyond scale separation

Published online by Cambridge University Press:  04 August 2021

Robert Altmann ,
Patrick Henning  and
Daniel Peterseim
Robert Altmann
Affiliation:
Institut für Mathematik, Universität Augsburg, 86159Augsburg, Germany E-mail: robert.altmann@math.uni-augsburg.de
Patrick Henning
Affiliation:
Fakultät für Mathematik, Ruhr-Universität Bochum, 44801Bochum, Germany E-mail: patrick.henning@ruhr-uni-bochum.de Department of Mathematics, KTH Royal Institute of Technology, SE-100 44 Stockholm, Sweden
Daniel Peterseim
Affiliation:
Institut für Mathematik, Universität Augsburg, 86159Augsburg, Germany E-mail: daniel.peterseim@math.uni-augsburg.de
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Numerical homogenization is a methodology for the computational solution of multiscale partial differential equations. It aims at reducing complex large-scale problems to simplified numerical models valid on some target scale of interest, thereby accounting for the impact of features on smaller scales that are otherwise not resolved. While constructive approaches in the mathematical theory of homogenization are restricted to problems with a clear scale separation, modern numerical homogenization methods can accurately handle problems with a continuum of scales. This paper reviews such approaches embedded in a historical context and provides a unified variational framework for their design and numerical analysis. Apart from prototypical elliptic model problems, the class of partial differential equations covered here includes wave scattering in heterogeneous media and serves as a template for more general multi-physics problems.

Type
Research Article
Information
Acta Numerica , Volume 30 , May 2021 , pp. 1 - 86
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

Footnotes

*

The work of D. Peterseim is part of a project that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement no. 865751).

References

Aarnes, J. E. (2004), On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation, Multiscale Model. Simul. 2, 421–439.CrossRefGoogle Scholar
Aarnes, J. E., Efendiev, Y. and Jiang, L. (2008), Mixed multiscale finite element methods using limited global information, Multiscale Model . Simul. 7, 655–676.Google Scholar
Abdulle, A. (2005), On a priori error analysis of fully discrete heterogeneous multiscale FEM, Multiscale Model . Simul. 4, 447–459.Google Scholar
Abdulle, A. (2009), The finite element heterogeneous multiscale method: A computational strategy for multiscale PDEs, in Multiple Scales Problems in Biomathematics, Mechanics, Physics and Numerics, Vol. 31 of GAKUTO International Series: Mathematical Sciences and Applications, Gakkōtosho, pp. 133–181.Google Scholar
Abdulle, A. and Bai, Y. (2012), Reduced basis finite element heterogeneous multiscale method for high-order discretizations of elliptic homogenization problems, J. Comput. Phys. 231, 7014–7036.CrossRefGoogle Scholar
Abdulle, A. and Budáč, O. (2015), An adaptive finite element heterogeneous multiscale method for Stokes flow in porous media, Multiscale Model . Simul. 13, 256–290.Google Scholar
Abdulle, A. and Henning, P. (2017), Localized orthogonal decomposition method for the wave equation with a continuum of scales, Math. Comp. 86, 549–587.CrossRefGoogle Scholar
Abdulle, A. and Pouchon, T. (2016), A priori error analysis of the finite element heterogeneous multiscale method for the wave equation over long time, SIAM J. Numer. Anal. 54, 1507–1534.CrossRefGoogle Scholar
Abdulle, A. and Pouchon, T. (2018), Effective models for long time wave propagation in locally periodic media, SIAM J. Numer. Anal. 56, 2701–2730.CrossRefGoogle Scholar
Abdulle, A., Arjmand, D. and Paganoni, E. (2019), Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems, C. R. Math. Acad. Sci. Paris 357, 545–551.CrossRefGoogle Scholar
Abdulle, A., E, W., Engquist, B. and Vanden-Eijnden, E. (2012), The heterogeneous multiscale method, in Acta Numerica, Vol. 21, Cambridge University Press, pp. 1–87.Google Scholar
Albrecht, F., Haasdonk, B., Kaulmann, S. and Ohlberger, M. (2012), The localized reduced basis multiscale method, in Algoritmy 2012: 19th Conference on Scientific Computing, Vysoké Tatry, Podbanské, Slovakia, Slovak University of Technology, Bratislava, pp. 393–403.Google Scholar
Allaire, G. (1992), Homogenization and two-scale convergence, SIAM J. Math. Anal. 23, 1482–1518.CrossRefGoogle Scholar
Allaire, G. and Briane, M. (1996), Multiscale convergence and reiterated homogenisation, Proc . Roy. Soc. Edinburgh Sect. A 126, 297–342.CrossRefGoogle Scholar
Allaire, G. and Brizzi, R. (2005), A multiscale finite element method for numerical homogenization, Multiscale Model . Simul. 4, 790–812.Google Scholar
Altmann, R. and Peterseim, D. (2019), Localized computation of eigenstates of random Schrödinger operators, SIAM J. Sci. Comput. 41, B1211–B1227.CrossRefGoogle Scholar
Altmann, R. and Verfürth, B. (2021), A multiscale method for heterogeneous bulk-surface coupling, Multiscale Model . Simul. 19, 374–400.Google Scholar
Altmann, R., Chung, E., Maier, R., Peterseim, D. and Pun, S.-M. (2020a), Computational multiscale methods for linear heterogeneous poroelasticity, J. Comput. Math. 38, 41–57.CrossRefGoogle Scholar
Altmann, R., Henning, P. and Peterseim, D. (2020b), Localization and delocalization of ground states of Bose–Einstein condensates under disorder. Available at arXiv:2006.00773.Google Scholar
Altmann, R., Henning, P. and Peterseim, D. (2020c), Quantitative Anderson localization of Schrödinger eigenstates under disorder potentials, Math. Models Methods Appl. Sci. 30, 917–955.CrossRefGoogle Scholar
Anderson, P. W. (1958), Absence of diffusion in certain random lattices, Phys. Rev. 109, 1492–1505.CrossRefGoogle Scholar
Araya, R., Harder, C., Paredes, D. and Valentin, F. (2013), Multiscale hybrid-mixed method, SIAM J. Numer. Anal. 51, 3505–3531.CrossRefGoogle Scholar
Arjmand, D. and Runborg, O. (2016), A time dependent approach for removing the cell boundary error in elliptic homogenization problems, J. Comput. Phys. 314, 206–227.CrossRefGoogle Scholar
Arjmand, D. and Runborg, O. (2017), Estimates for the upscaling error in heterogeneous multiscale methods for wave propagation problems in locally periodic media, Multiscale Model . Simul. 15, 948–976.Google Scholar
Arjmand, D. and Stohrer, C. (2016), A finite element heterogeneous multiscale method with improved control over the modeling error, Commun . Math. Sci. 14, 463–487.Google Scholar
Babuška, I. (1971), Error-bounds for finite element method, Numer . Math. 16, 322–333.Google Scholar
Babuška, I. (1976), Homogenization and its application: Mathematical and computational problems, in Numerical Solution of Partial Differential Equations III (SYNSPADE), Academic Press, pp. 89–116.CrossRefGoogle Scholar
Babuška, I. and Lipton, R. (2011), Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model . Simul. 9, 373–406.Google Scholar
Babuška, I. and Osborn, J. E. (1983), Generalized finite element methods: Their performance and their relation to mixed methods, SIAM J. Numer. Anal. 20, 510–536.CrossRefGoogle Scholar
Babuška, I. and Osborn, J. E. (2000), Can a finite element method perform arbitrarily badly?, Math. Comp. 69, 443–462.CrossRefGoogle Scholar
Babuška, I. and Sauter, S. A. (1997), Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers?, SIAM J. Numer. Anal. 34, 2392–2423.CrossRefGoogle Scholar
Babuška, I., Caloz, G. and Osborn, J. E. (1994), Special finite element methods for a class of second order elliptic problems with rough coefficients, SIAM J. Numer. Anal. 31, 945–981.CrossRefGoogle Scholar
Babuška, I., Lipton, R., Sinz, P. and Stuebner, M. (2020), Multiscale-spectral GFEM and optimal oversampling, Comput . Methods Appl. Mech. Engrg 364, 112960.CrossRefGoogle Scholar
Baker, G. A. (1976), Error estimates for finite element methods for second order hyperbolic equations, SIAM J. Numer. Anal. 13, 564–576.CrossRefGoogle Scholar
Bastian, P., Blatt, M., Dedner, A., Dreier, N.-A., Engwer, C., Fritze, R., Gräser, C., Grüninger, C., Kempf, D., Klöfkorn, R., Ohlberger, M. and Sander, O. (2021), The Dune framework: Basic concepts and recent developments, Comput. Math. Appl. 81, 75–112.CrossRefGoogle Scholar
Bebendorf, M. (2003), A note on the Poincaré inequality for convex domains, Z . Anal. Anwendungen 22, 751–756.CrossRefGoogle Scholar
Bensoussan, A., Lions, J.-L. and Papanicolaou, G. (1978), Asymptotic Analysis for Periodic Structures, North-Holland.Google Scholar
Billy, J., Josse, V., Zuo, Z., Bernard, A., Hambrecht, B., Lugan, P., Clément, D., Sanchez-Palencia, L., Bouyer, P. and Aspect, A. (2008), Direct observation of Anderson localization of matter waves in a controlled disorder, Nature 453, 891–894.CrossRefGoogle Scholar
Boffi, D., Brezzi, F. and Fortin, M. (2008), Finite elements for the Stokes problem, in Mixed Finite Elements, Compatibility Conditions, and Applications, Vol. 1939 of Lecture Notes in Mathematics, Springer, pp. 45–100.CrossRefGoogle Scholar
Braess, D. (2007), Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory, third edition, Cambridge University Press.CrossRefGoogle Scholar
Brenner, S. C. (1994), Two-level additive Schwarz preconditioners for nonconforming finite elements, in Domain Decomposition Methods in Scientific and Engineering Computing (University Park, PA, 1993), Vol. 180 of Contemporary Mathematics, American Mathematical Society, pp. 9–14.CrossRefGoogle Scholar
Brezzi, F. and Fortin, M. (1991), Mixed and Hybrid Finite Element Methods, Vol. 15 of Springer Series in Computational Mathematics, Springer.CrossRefGoogle Scholar
Brezzi, F. and Russo, A. (1994), Choosing bubbles for advection–diffusion problems, Math. Models Methods Appl. Sci. 4, 571–587.CrossRefGoogle Scholar
Brezzi, F., Bristeau, M. O., Franca, L. P., Mallet, M. and Rogé, G. (1992), A relationship between stabilized finite element methods and the Galerkin method with bubble functions, Comput. Methods Appl. Mech. Engrg 96, 117–129.CrossRefGoogle Scholar
Brezzi, F., Franca, L. P., Hughes, T. J. R. and Russo, A. (1997), b = ∫ G, Comput. Methods Appl. Mech. Engrg 145, 329–339.CrossRefGoogle Scholar
Brown, D. L. and Gallistl, D. (2016), Multiscale sub-grid correction method for time-harmonic high-frequency elastodynamics with wavenumber explicit bounds. Available at arXiv:1608.04243.Google Scholar
Brown, D. L. and Peterseim, D. (2016), A multiscale method for porous microstructures, Multiscale Model . Simul. 14, 1123–1152.Google Scholar
Brown, D. L., Gallistl, D. and Peterseim, D. (2017), Multiscale Petrov–Galerkin method for high-frequency heterogeneous Helmholtz equations, in Meshfree Methods for Partial Differential Equations VIII, Vol. 115 of Lecture Notes in Computer Science and Engineering, Springer, pp. 85–115.CrossRefGoogle Scholar
Brown, D. L., Gedicke, J. and Peterseim, D. (2018), Numerical homogenization of heterogeneous fractional Laplacians, Multiscale Model . Simul. 16, 1305–1332.Google Scholar
Buhr, A. and Smetana, K. (2018), Randomized local model order reduction, SIAM J. Sci. Comput. 40, A2120–A2151.CrossRefGoogle Scholar
Buhr, A., Iapichino, L., Ohlberger, M., Rave, S., Schindler, F. and Smetana, K. (2021), Localized model reduction for parameterized problems, in Snapshot-Based Methods and Algorithms (Benner, P. et al., eds), Vol. 3 of Model Order Reduction, De Gruyter.Google Scholar
Caiazzo, A., Maier, R. and Peterseim, D. (2020), Reconstruction of quasi-local numerical effective models from low-resolution measurements, J. Sci. Comput. 85, 10.CrossRefGoogle Scholar
Calo, V. M., Efendiev, Y., Galvis, J. and Li, G. (2016), Randomized oversampling for generalized multiscale finite element methods, Multiscale Model . Simul. 14, 482–501.Google Scholar
Chamoin, L. and Legoll, F. (2018), A posteriori error estimation and adaptive strategy for the control of MsFEM computations, Comput . Methods Appl. Mech. Engrg 336, 1–38.CrossRefGoogle Scholar
Chaumont-Frelet, T. and Verfürth, B. (2020), A generalized finite element method for problems with sign-changing coefficients. Available at arXiv:2002.10818.Google Scholar
Chaumont-Frelet, T., Gallistl, D., Nicaise, S. and Tomezyk, J. (2018), Wavenumber explicit convergence analysis for finite element discretizations of time-harmonic wave propagation problems with perfectly matched layers. Available at hal-01887267.Google Scholar
Chen, K., Li, Q., Lu, J. and Wright, S. J. (2020), Random sampling and efficient algorithms for multiscale PDEs, SIAM J. Sci. Comput. 42, A2974–A3005.CrossRefGoogle Scholar
Chertock, A. and Levy, D. (2005), On wavelet-based numerical homogenization, Multiscale Model . Simul. 3, 65–88.Google Scholar
Cheung, S. W., Chung, E. T., Efendiev, Y., Leung, W. T. and Vasilyeva, M. (2020), Constraint energy minimizing generalized multiscale finite element method for dual continuum model, Commun . Math. Sci. 18, 663–685.Google Scholar
Chu, V. T. and Hoang, V. H. (2018), High-dimensional finite elements for multiscale Maxwell-type equations, IMA J. Numer. Anal. 38, 227–270.CrossRefGoogle Scholar
Chung, E. T. and Li, Y. (2019), Adaptive generalized multiscale finite element methods for H(curl)-elliptic problems with heterogeneous coefficients, J. Comput. Appl. Math. 345, 357–373.CrossRefGoogle Scholar
Chung, E. T., Efendiev, Y. and Fu, S. (2014), Generalized multiscale finite element method for elasticity equations, GEM Int. J. Geomath. 5, 225–254.CrossRefGoogle Scholar
Chung, E. T., Efendiev, Y. and Leung, W. T. (2017), An online generalized multiscale discontinuous Galerkin method (GMsDGM) for flows in heterogeneous media, Commun . Comput. Phys. 21, 401–422.CrossRefGoogle Scholar
Chung, E. T., Efendiev, Y. and Leung, W. T. (2019), Generalized multiscale finite element methods with energy minimizing oversampling, Internat. J. Numer. Methods Engrg 117, 316–343.CrossRefGoogle Scholar
Chung, E. T., Efendiev, Y., Li, G. and Vasilyeva, M. (2016), Generalized multiscale finite element methods for problems in perforated heterogeneous domains, Appl. Anal. 95, 2254–2279.CrossRefGoogle Scholar
Cicuttin, M., Ern, A. and Lemaire, S. (2019), A hybrid high-order method for highly oscillatory elliptic problems, Comput . Methods Appl. Math. 19, 723–748.Google Scholar
Cioranescu, D. and Donato, P. (1999), An Introduction to Homogenization, Vol. 17 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press.Google Scholar
Giorgi, E. De (1975), Sulla convergenza di alcune successioni d’integrali del tipo dell’area, Rend . Mat. (6) 8, 277–294.Google Scholar
Demkowicz, L. (2006), Babuška ⇔ Brezzi?? Report, Texas Institute for Computational and Applied Mathematics, University of Texas at Austin.Google Scholar
Pietro, D. A. Di and Ern, A. (2012), Mathematical Aspects of Discontinuous Galerkin Methods, Springer.CrossRefGoogle Scholar
Dong, H. and Kim, S. (2009), Green’s matrices of second order elliptic systems with measurable coefficients in two dimensional domains, Trans. Amer. Math. Soc. 361, 3303–3323.CrossRefGoogle Scholar
Dorobantu, M. and Engquist, B. (1998), Wavelet-based numerical homogenization, SIAM J. Numer. Anal. 35, 540–559.CrossRefGoogle Scholar
W, E and Engquist, B. (2003), The heterogeneous multiscale methods, Commun . Math. Sci. 1, 87–132.Google Scholar
Efendiev, Y. and Hou, T. Y. (2009), Multiscale Finite Element Methods : Theory and Applications, Vol. 4 of Surveys and Tutorials in the Applied Mathematical Sciences, Springer.Google Scholar
Efendiev, Y., Galvis, J. and Hou, T. Y. (2013), Generalized multiscale finite element methods (GMsFEM), J. Comput. Physics 251, 116–135.CrossRefGoogle Scholar
Efendiev, Y., Hou, T. Y. and Wu, X.-H. (2000), Convergence of a nonconforming multiscale finite element method, SIAM J. Numer. Anal. 37, 888–910.CrossRefGoogle Scholar
Efros, A. L. and Pokrovsky, A. L. (2004), Dielectric photonic crystal as medium with negative electric permittivity and magnetic permeability, Solid State Commun. 129, 643–647.CrossRefGoogle Scholar
Elfverson, D., Georgoulis, E. H. and Målqvist, A. (2013a), An adaptive discontinuous Galerkin multiscale method for elliptic problems, Multiscale Model . Simul. 11, 747– 765.Google Scholar
Elfverson, D., Georgoulis, E. H., Målqvist, A. and Peterseim, D. (2013b), Convergence of a discontinuous Galerkin multiscale method, SIAM J. Numer. Anal. 51, 3351–3372.CrossRefGoogle Scholar
Elfverson, D., Ginting, V. and Henning, P. (2015), On multiscale methods in Petrov–Galerkin formulation, Numer . Math. 131, 643–682.Google Scholar
Elfverson, D., Larson, M. G. and Målqvist, A. (2017), Multiscale methods for problems with complex geometry, Comput . Methods Appl. Mech. Engrg 321, 103–123.CrossRefGoogle Scholar
Engquist, B. (1987), Computation of oscillatory solutions to partial differential equations, in Nonlinear Hyperbolic Problems (Carasso, C., Serre, D. and Raviart, P.-A., eds), Vol. 1270 of Lecture Notes in Mathematics, Springer, pp. 10–22.CrossRefGoogle Scholar
Engquist, B. and Runborg, O. (2002), Wavelet-based numerical homogenization with applications, in Multiscale and Multiresolution Methods, Vol. 20 of Lecture Notes in Computer Science and Engineering, Springer, pp. 97–148.CrossRefGoogle Scholar
Engwer, C., Henning, P., Målqvist, A. and Peterseim, D. (2019), Efficient implementation of the localized orthogonal decomposition method, Comput . Methods Appl. Mech. Engrg 350, 123–153.CrossRefGoogle Scholar
Ern, A. and Guermond, J.-L. (2017), Finite element quasi-interpolation and best approximation, ESAIM Math. Model. Numer. Anal. 51, 1367–1385.CrossRefGoogle Scholar
Esterhazy, S. and Melenk, J. M. (2012), On stability of discretizations of the Helmholtz equation, in Numerical Analysis of Multiscale Problems, Vol. 83 of Lecture Notes in Computer Science and Engineering, Springer, pp. 285–324.CrossRefGoogle Scholar
Evans, L. C. (2010), Partial Differential Equations , Vol. 19 of Graduate Studies in Mathematics, second edition, American Mathematical Society.Google Scholar
Falk, R. S. and Winther, R. (2014), Local bounded cochain projections, Math. Comp. 83, 2631–2656.CrossRefGoogle Scholar
Falk, R. S. and Winther, R. (2015), Double complexes and local cochain projections, Numer . Methods Partial Differential Equations 31, 541–551.CrossRefGoogle Scholar
Feischl, M. and Peterseim, D. (2020), Sparse compression of expected solution operators, SIAM J. Numer. Anal. 58, 3144–3164.CrossRefGoogle Scholar
Fernandes, P. and Raffetto, M. (2005), Existence, uniqueness and finite element approximation of the solution of time-harmonic electromagnetic boundary value problems involving metamaterials, COMPEL 24, 1450–1469.CrossRefGoogle Scholar
Fischer, J., Gallistl, D. and Peterseim, D. (2021), A priori error analysis of a numerical stochastic homogenization method, SIAM J. Numer. Anal. 59, 660–674.CrossRefGoogle Scholar
Franca, L. P. and Russo, A. (1996), Deriving upwinding, mass lumping and selective reduced integration by residual-free bubbles, Appl. Math. Lett. 9, 83–88.CrossRefGoogle Scholar
Fu, S., Altmann, R., Chung, E., Maier, R., Peterseim, D. and Pun, S.-M. (2019), Computational multiscale methods for linear poroelasticity with high contrast, J. Comput. Phys. 395, 286–297.CrossRefGoogle Scholar
Gallistl, D. and Peterseim, D. (2015), Stable multiscale Petrov–Galerkin finite element method for high frequency acoustic scattering, Comput . Methods Appl. Mech. Engrg 295, 1–17.CrossRefGoogle Scholar
Gallistl, D. and Peterseim, D. (2017), Computation of quasi-local effective diffusion tensors and connections to the mathematical theory of homogenization, Multiscale Model . Simul. 15, 1530–1552.Google Scholar
Gallistl, D. and Peterseim, D. (2019), Numerical stochastic homogenization by quasilocal effective diffusion tensors, Commun . Math. Sci. 17, 637–651.Google Scholar
Gallistl, D., Henning, P. and Verfürth, B. (2018), Numerical homogenization of H(curl)-problems, SIAM J. Numer. Anal. 56, 1570–1596.CrossRefGoogle Scholar
Gao, K., Fu, S., Gibson, R. L., Chung, E. T. and Efendiev, Y. (2015), Generalized multiscale finite-element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, J. Comput. Phys. 295, 161–188.CrossRefGoogle Scholar
Gloria, A. (2006), An analytical framework for the numerical homogenization of monotone elliptic operators and quasiconvex energies, Multiscale Model . Simul. 5, 996–1043.Google Scholar
Gloria, A. (2008), An analytical framework for numerical homogenization II: Windowing and oversampling, Multiscale Model . Simul. 7, 274–293.Google Scholar
Gloria, A. (2012), Numerical homogenization: Survey, new results, and perspectives, ESAIM Proc. 37, 50–116.CrossRefGoogle Scholar
Graham, I. G., Hackbusch, W. and Sauter, S. A. (2005), Finite elements on degenerate meshes: Inverse-type inequalities and applications, IMA J. Numer. Anal. 25, 379–407.CrossRefGoogle Scholar
Grasedyck, L., Greff, I. and Sauter, S. (2012), The AL basis for the solution of elliptic problems in heterogeneous media, Multiscale Model . Simul. 10, 245–258.Google Scholar
Grüter, M. and Widman, K.-O. (1982), The Green function for uniformly elliptic equations, Manuscripta Math. 37, 303–342.CrossRefGoogle Scholar
Hackbusch, W. (2015), Hierarchical Matrices: Algorithms and Analysis, Vol. 49 of Springer Series in Computational Mathematics, Springer.CrossRefGoogle Scholar
Harbrecht, H. and Schwab, C. (2011), Sparse tensor finite elements for elliptic multiple scale problems, Comput . Methods Appl. Mech. Engrg 200, 3100–3110.CrossRefGoogle Scholar
Heinlein, A., Hetmaniuk, U., Klawonn, A. and Rheinbach, O. (2015), The approximate component mode synthesis special finite element method in two dimensions: Parallel implementation and numerical results, J. Comput. Appl. Math. 289, 116–133.CrossRefGoogle Scholar
Hellman, F. (2017), Gridlod, https://github.com/fredrikhellman/gridlod.Google Scholar
Hellman, F. and Målqvist, A. (2017), Contrast independent localization of multiscale problems, Multiscale Model . Simul. 15, 1325–1355.Google Scholar
Hellman, F. and Målqvist, A. (2019), Numerical homogenization of elliptic PDEs with similar coefficients, Multiscale Model . Simul. 17, 650–674.Google Scholar
Hellman, F., Henning, P. and Målqvist, A. (2016), Multiscale mixed finite elements, Discrete Contin . Dyn. Syst. Ser. S 9, 1269–1298.Google Scholar
Hellman, F., Keil, T. and Målqvist, A. (2020), Numerical upscaling of perturbed diffusion problems, SIAM J. Sci. Comput. 42, A2014–A2036.CrossRefGoogle Scholar
Helmig, R. (1997), Multiphase Flow and Transport Processes in the Subsurface: A Contribution to the Modeling of Hydrosystems, Springer.CrossRefGoogle Scholar
Hennig, P., Maier, R., Peterseim, D., Schillinger, D., Verfürth, B. and Kästner, M. (2020), A diffuse modeling approach for embedded interfaces in linear elasticity, GAMM-Mitt. 43, e202000001.CrossRefGoogle Scholar
Henning, P. and Målqvist, A. (2014), Localized orthogonal decomposition techniques for boundary value problems, SIAM J. Sci. Comput. 36, A1609–A1634.CrossRefGoogle Scholar
Henning, P. and Ohlberger, M. (2009), The heterogeneous multiscale finite element method for elliptic homogenization problems in perforated domains, Numer . Math. 113, 601– 629.Google Scholar
Henning, P. and Persson, A. (2016), A multiscale method for linear elasticity reducing Poisson locking, Comput . Methods Appl. Mech. Engrg 310, 156–171.CrossRefGoogle Scholar
Henning, P. and Persson, A. (2020), Computational homogenization of time-harmonic Maxwell’s equations, SIAM J. Sci. Comput. 42, B581–B607.CrossRefGoogle Scholar
Henning, P. and Peterseim, D. (2013), Oversampling for the multiscale finite element method, Multiscale Model . Simul. 11, 1149–1175.Google Scholar
Henning, P. and Wärnegård, J. (2020), Superconvergence of time invariants for the Gross– Pitaevskii equation. Available at arXiv:2008.07575.Google Scholar
Henning, P., Målqvist, A. and Peterseim, D. (2014a), A localized orthogonal decomposition method for semi-linear elliptic problems, ESAIM Math. Model. Numer. Anal. 48, 1331– 1349.CrossRefGoogle Scholar
Henning, P., Målqvist, A. and Peterseim, D. (2014b), Two-level discretization techniques for ground state computations of Bose–Einstein condensates, SIAM J. Numer. Anal. 52, 1525–1550.CrossRefGoogle Scholar
Henning, P., Morgenstern, P. and Peterseim, D. (2015), Multiscale partition of unity, in Meshfree Methods for Partial Differential Equations VII (Griebel, M. and Schweitzer, M. A., eds), Vol. 100 of Lecture Notes in Computational Science and Engineering, Springer, pp. 185–204.CrossRefGoogle Scholar
Henning, P., Ohlberger, M. and Schweizer, B. (2014c), An adaptive multiscale finite element method, SIAM Multiscale Model . Simul. 12, 1078–1107.Google Scholar
Henning, P., Ohlberger, M. and Verfürth, B. (2016), A new heterogeneous multiscale method for time-harmonic Maxwell’s equations, SIAM J. Numer. Anal. 54, 3493–3522.CrossRefGoogle Scholar
Hesthaven, J. S., Zhang, S. and Zhu, X. (2014), High-order multiscale finite element method for elliptic problems, Multiscale Model . Simul. 12, 650–666.Google Scholar
Hetmaniuk, U. and Klawonn, A. (2014), Error estimates for a two-dimensional special finite element method based on component mode synthesis, Electron . Trans. Numer. Anal. 41, 109–132.Google Scholar
Hetmaniuk, U. and Lehoucq, R. B. (2010), A special finite element method based on component mode synthesis, ESAIM Math. Model. Numer. Anal. 44, 401–420.CrossRefGoogle Scholar
Hiptmair, R. (2002), Finite elements in computational electromagnetism, in Acta Numerica, Vol. 11, Cambridge University Press, pp. 237–339.Google Scholar
Hiptmair, R. and Xu, J. (2007), Nodal auxiliary space preconditioning in H(curl) and H(div) spaces, SIAM J. Numer. Anal. 45, 2483–2509.CrossRefGoogle Scholar
Hoang, V. H. (2008), Sparse finite element method for periodic multiscale nonlinear monotone problems, Multiscale Model . Simul. 7, 1042–1072.Google Scholar
Hoang, V. H. and Schwab, C. (2005), High-dimensional finite elements for elliptic problems with multiple scales, Multiscale Model . Simul. 3, 168–194.Google Scholar
Hochbruck, M., Maier, B. and Stohrer, C. (2019), Heterogeneous multiscale method for Maxwell’s equations, Multiscale Model . Simul. 17, 1147–1171.Google Scholar
Hou, T. Y. and Wu, X.-H. (1997), A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134, 169–189.CrossRefGoogle Scholar
Hou, T. Y. and Zhang, P. (2017), Sparse operator compression of higher-order elliptic operators with rough coefficients, Res . Math. Sci. 4, 24.Google Scholar
Hou, T. Y., Huang, D., Lam, K. C. and Zhang, P. (2018), An adaptive fast solver for a general class of positive definite matrices via energy decomposition, Multiscale Model . Simul. 16, 615–678.Google Scholar
Hou, T. Y., Li, Q. and Zhang, P. (2017a), Exploring the locally low dimensional structure in solving random elliptic PDEs, Multiscale Model . Simul. 15, 661–695.Google Scholar
Hou, T. Y., Li, Q. and Zhang, P. (2017b), A sparse decomposition of low rank symmetric positive semidefinite matrices, Multiscale Model . Simul. 15, 410–444.Google Scholar
Hou, T. Y., Wu, X.-H. and Cai, Z. (1999), Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients, Math. Comp. 68, 913–943.CrossRefGoogle Scholar
Hou, T. Y., Wu, X.-H. and Zhang, Y. (2004), Removing the cell resonance error in the multiscale finite element method via a Petrov–Galerkin formulation, Commun . Math. Sci. 2, 185–205.Google Scholar
Hu, H., Strybulevych, A., Page, J. H., Skipetrov, S. E. and Tiggelen, B. A. (2008), Localization of ultrasound in a three-dimensional elastic network, Nat. Phys. 4, 945– 948.CrossRefGoogle Scholar
Hughes, T. J. R. (1995), Multiscale phenomena: Green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. Methods Appl. Mech. Engrg 127, 387–401.CrossRefGoogle Scholar
Hughes, T. J. R. and Sangalli, G. (2007), Variational multiscale analysis: The fine-scale Green’s function, projection, optimization, localization, and stabilized methods, SIAM J. Numer. Anal. 45, 539–557.CrossRefGoogle Scholar
Hughes, T. J. R., Feijóo, G. R., Mazzei, L. and Quincy, J.-B. (1998), The variational multiscale method: A paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg 166, 3–24.CrossRefGoogle Scholar
Jiang, L. and Efendiev, Y. (2012), A priori estimates for two multiscale finite element methods using multiple global fields to wave equations, Numer . Methods Partial Differential Equations 28, 1869–1892.CrossRefGoogle Scholar
Jiang, L., Efendiev, Y. and Ginting, V. (2007), Multiscale methods for parabolic equations with continuum spatial scales, Discrete Contin . Dyn. Syst. Ser. B 8, 833–859.Google Scholar
Jiang, L., Efendiev, Y. and Ginting, V. (2010), Analysis of global multiscale finite element methods for wave equations with continuum spatial scales, Appl. Numer. Math. 60, 862–876.CrossRefGoogle Scholar
Jikov, V. V., Kozlov, S. M. and Oleinik, O. A. (1994), Homogenization of Differential Operators and Integral Functionals, Springer.CrossRefGoogle Scholar
Kettil, G., Målqvist, A., Mark, A., Fredlund, M., Wester, K. and Edelvik, F. (2020), Numerical upscaling of discrete network models, BIT 60, 67–92.CrossRefGoogle Scholar
Klawonn, A., Kühn, M. and Rheinbach, O. (2018), Adaptive FETI-DP and BDDC methods with a generalized transformation of basis for heterogeneous problems, Electron . Trans. Numer. Anal. 49, 1–27.Google Scholar
Klawonn, A., Radtke, P. and Rheinbach, O. (2015), FETI-DP methods with an adaptive coarse space, SIAM J. Numer. Anal. 53, 297–320.CrossRefGoogle Scholar
Kornhuber, R. and Yserentant, H. (2016), Numerical homogenization of elliptic multiscale problems by subspace decomposition, Multiscale Model . Simul. 14, 1017–1036.Google Scholar
Kornhuber, R., Peterseim, D. and Yserentant, H. (2018), An analysis of a class of variational multiscale methods based on subspace decomposition, Math. Comp. 87, 2765–2774.CrossRefGoogle Scholar
Lamacz, A. and Schweizer, B. (2013), Effective Maxwell equations in a geometry with flat rings of arbitrary shape, SIAM J. Math. Anal. 45, 1460–1494.CrossRefGoogle Scholar
Lamacz, A. and Schweizer, B. (2016), A negative index meta-material for Maxwell’s equations, SIAM J. Math. Anal. 48, 4155–4174.CrossRefGoogle Scholar
Larson, M. G. and Målqvist, A. (2005), Adaptive variational multiscale methods based on a posteriori error estimation: Duality techniques for elliptic problems, in Multiscale Methods in Science and Engineering, Vol. 44 of Lecture Notes in Computer Science and Engineering, Springer, pp. 181–193.CrossRefGoogle Scholar
Larson, M. G. and Målqvist, A. (2007), Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems, Comput. Methods Appl. Mech. Engrg 196, 2313–2324.CrossRefGoogle Scholar
Larson, M. G. and Målqvist, A. (2009), An adaptive variational multiscale method for convection–diffusion problems, Comm. Numer. Methods Engrg 25, 65–79.CrossRefGoogle Scholar
Bris, C. Le, Legoll, F. and Lozinski, A. (2013), MsFEM à la Crouzeix–Raviart for highly oscillatory elliptic problems, Chin . Ann. Math. Ser. B 34, 113–138.CrossRefGoogle Scholar
Bris, C. Le, Legoll, F. and Lozinski, A. (2014a), An MsFEM type approach for perforated domains, Multiscale Model . Simul. 12, 1046–1077.Google Scholar
Bris, C. Le, Legoll, F. and Thomines, F. (2014b), Multiscale finite element approach for ‘weakly’ random problems and related issues, ESAIM Math. Model. Numer. Anal. 48, 815–858.CrossRefGoogle Scholar
Li, G., Peterseim, D. and Schedensack, M. (2018), Error analysis of a variational multiscale stabilization for convection-dominated diffusion equations in two dimensions, IMA J. Numer. Anal. 38, 1229–1253.CrossRefGoogle Scholar
Lipton, R. and Schweizer, B. (2018), Effective Maxwell’s equations for perfectly conducting split ring resonators, Arch . Ration. Mech. Anal. 229, 1197–1221.CrossRefGoogle Scholar
Maier, R. (2020), Computational multiscale methods in unstructured heterogeneous media. PhD thesis, University of Augsburg.Google Scholar
Maier, R. (2021), A high-order approach to elliptic multiscale problems with general unstructured coefficients, SIAM J. Numer. Anal. 59, 1067–1089.CrossRefGoogle Scholar
Maier, R. and Peterseim, D. (2019), Explicit computational wave propagation in micro-heterogeneous media, BIT 59, 443–462.CrossRefGoogle Scholar
Maier, R. and Verfürth, B. (2020), Multiscale scattering in nonlinear Kerr-type media. Available at arXiv:2011.09168.Google Scholar
Målqvist, A. (2005), Adaptive variational multiscale methods. PhD thesis, Chalmers University of Technology.Google Scholar
Målqvist, A. (2011), Multiscale methods for elliptic problems, Multiscale Model . Simul. 9, 1064–1086.Google Scholar
Målqvist, A. and Persson, A. (2017), A generalized finite element method for linear ther-moelasticity, ESAIM Math. Model. Numer. Anal. 51, 1145–1171.CrossRefGoogle Scholar
Målqvist, A. and Persson, A. (2018), Multiscale techniques for parabolic equations, Numer . Math. 138, 191–217.Google Scholar
Målqvist, A. and Peterseim, D. (2014), Localization of elliptic multiscale problems, Math. Comp. 83, 2583–2603.CrossRefGoogle Scholar
Målqvist, A. and Peterseim, D. (2015), Computation of eigenvalues by numerical upscaling, Numer . Math. 130, 337–361.Google Scholar
Målqvist, A. and Peterseim, D. (2017), Generalized finite element methods for quadratic eigenvalue problems, ESAIM Math. Model. Numer. Anal. 51, 147–163.CrossRefGoogle Scholar
Målqvist, A. and Peterseim, D. (2020), Numerical Homogenization by Localized Orthogonal Decomposition, Society for Industrial and Applied Mathematics (SIAM).CrossRefGoogle Scholar
Matache, A.-M. and Schwab, C. (2002), Two-scale FEM for homogenization problems, ESAIM Math. Model. Numer. Anal. 36, 537–572.CrossRefGoogle Scholar
Melenk, J. M. (1995), On generalized finite-element methods. PhD thesis, University of Maryland, College Park.Google Scholar
Melenk, J. M. and Sauter, S. (2011), Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation, SIAM J. Numer. Anal. 49, 1210–1243.CrossRefGoogle Scholar
Monk, P. (2003), Finite Element Methods for Maxwell’s Equations, Numerical Mathematics and Scientific Computation, Oxford University Press.CrossRefGoogle Scholar
Murat, F. and Tartar, L. (1978), H-convergence, Séminaire d’Analyse Fonctionnelle et Numérique de l’Université d’Alger . Reprinted in Topics in the Mathematical Modelling of Composite Materials (Cherkaev, A. and Kohn, R., eds), Birkhäuser (1997), pp. 21–43.Google Scholar
Niessner, J. and Helmig, R. (2006), Multi-scale modelling of two-phase–two-component processes in heterogeneous porous media, Numer . Linear Algebra Appl. 13, 699–715.CrossRefGoogle Scholar
Nolen, J., Papanicolaou, G. and Pironneau, O. (2008), A framework for adaptive multiscale methods for elliptic problems, Multiscale Model . Simul. 7, 171–196.Google Scholar
Ohlberger, M. (2005), A posteriori error estimates for the heterogeneous multiscale finite element method for elliptic homogenization problems, Multiscale Model . Simul. 4, 88– 114.Google Scholar
Ohlberger, M. and Schindler, F. (2014), A-posteriori error estimates for the localized reduced basis multi-scale method, in Finite Volumes for Complex Applications VII: Methods and Theoretical Aspects, Vol. 77 of Springer Proceedings in Mathematics & Statistics, Springer, pp. 421–429.Google Scholar
Ohlberger, M. and Schindler, F. (2015), Error control for the localized reduced basis multiscale method with adaptive on-line enrichment, SIAM J. Sci. Comput. 37, A2865– A2895.CrossRefGoogle Scholar
Oswald, P. (1993), On a BPX-preconditioner for P1 elements, Computing 51, 125–133.CrossRefGoogle Scholar
Owhadi, H. (2015), Bayesian numerical homogenization, Multiscale Model . Simul. 13, 812–828.Google Scholar
Owhadi, H. (2017), Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games, SIAM Rev. 59, 99–149.CrossRefGoogle Scholar
Owhadi, H. and Scovel, C. (2019), Operator-Adapted Wavelets, Fast Solvers, and Numerical Homogenization: From a Game Theoretic Approach to Numerical Approximation and Algorithm Design, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press.CrossRefGoogle Scholar
Owhadi, H. and Zhang, L. (2007), Metric-based upscaling, Comm . Pure Appl. Math. 60, 675–723.CrossRefGoogle Scholar
Owhadi, H. and Zhang, L. (2011), Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast, Multiscale Model . Simul. 9, 1373–1398.Google Scholar
Owhadi, H. and Zhang, L. (2017), Gamblets for opening the complexity-bottleneck of implicit schemes for hyperbolic and parabolic ODEs/PDEs with rough coefficients, J. Comput. Phys. 347, 99–128.CrossRefGoogle Scholar
Owhadi, H., Zhang, L. and Berlyand, L. (2014), Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM Math. Model. Numer. Anal. 48, 517–552.CrossRefGoogle Scholar
Payne, L. E. and Weinberger, H. F. (1960), An optimal Poincaré inequality for convex domains, Arch . Rational Mech. Anal. 5, 286–292.CrossRefGoogle Scholar
Pechstein, C., Sarkis, M. and Scheichl, R. (2013), New theoretical coefficient robustness results for FETI-DP, in Domain Decomposition Methods in Science and Engineering, Vol. 91 of Lecture Notes in Computer Science and Engineering, Springer, pp. 313–320.CrossRefGoogle Scholar
Pendry, J. B. (2000), Negative refraction makes a perfect lens, Phys. Rev. Lett. 85, 3966– 3969.CrossRefGoogle ScholarPubMed
Pendry, J. B., Schurig, D. and Smith, D. R. (2006), Controlling electromagnetic fields, Science 312, 1780–1782.CrossRefGoogle ScholarPubMed
Peterseim, D. (2016), Variational multiscale stabilization and the exponential decay of fine-scale correctors, in Building Bridges: Connections and Challenges in Modern Approaches to Numerical Partial Differential Equations, Vol. 114 of Lecture Notes in Computational Science and Engineering, Springer, pp. 341–367.Google Scholar
Peterseim, D. (2017), Eliminating the pollution effect in Helmholtz problems by local subscale correction, Math. Comp. 86, 1005–1036.CrossRefGoogle Scholar
Peterseim, D. and Sauter, S. (2012), Finite elements for elliptic problems with highly varying, nonperiodic diffusion matrix, Multiscale Model . Simul. 10, 665–695.Google Scholar
Peterseim, D. and Schedensack, M. (2017), Relaxing the CFL condition for the wave equation on adaptive meshes, J. Sci. Comput. 72, 1196–1213.CrossRefGoogle Scholar
Peterseim, D. and Scheichl, R. (2016), Robust numerical upscaling of elliptic multiscale problems at high contrast, Comput . Methods Appl. Math. 16, 579–603.Google Scholar
Peterseim, D. and Verfürth, B. (2020), Computational high frequency scattering from high-contrast heterogeneous media, Math. Comp. 89, 2649–2674.CrossRefGoogle Scholar
Peterseim, D., Varga, D. and Verfürth, B. (2020), From domain decomposition to homogenization theory, in Domain Decomposition Methods in Science and Engineering XXV (Haynes, R. et al., ed.), Vol. 138 of Lecture Notes in Computer Science and Engineering, Springer, pp. 29–40.CrossRefGoogle Scholar
Pinkus, A. (1985), n-Widths in Approximation Theory, Springer.CrossRefGoogle Scholar
Pokrovsky, A. L. and Efros, A. L. (2003), Diffraction theory and focusing of light by a slab of left-handed material, Modern Phys. Lett. B 338, 333–337.Google Scholar
Roati, G., D’Errico, C., Fallani, L., Fattori, M., Fort, C., Zaccanti, M., Modugno, G., Modugno, M. and Inguscio, M. (2008), Anderson localization of a non-interacting Bose–Einstein condensate, Nature 453, 895–898.CrossRefGoogle ScholarPubMed
Schäfer, F., Katzfuss, M. and Owhadi, H. (2020), Sparse Cholesky factorization by Kullback– Leibler minimization. Available at arXiv:2004.14455 (to appear in SIAM Multiscale Model. Simul.).CrossRefGoogle Scholar
Schäfer, F., Sullivan, T. J. and Owhadi, H. (2021), Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexity, Multiscale Model . Simul. 19, 688–730.Google Scholar
Scheid, R. E. Jr (1983), The accurate numerical solution of highly oscillatory ordinary differential equations, Math. Comp. 41, 487–509.CrossRefGoogle Scholar
Schwab, C. (1998), p- and hp- Finite Element Methods: Theory and Applications in Solid and Fluid Mechanics, Numerical Mathematics and Scientific Computation, Clarendon Press.Google Scholar
Smetana, K. and Patera, A. T. (2016), Optimal local approximation spaces for component-based static condensation procedures, SIAM J. Sci. Comput. 38, A3318–A3356.CrossRefGoogle Scholar
Spagnolo, S. (1968), Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche, Ann . Sc. Norm. Super. Pisa Cl. Sci. 22, 571–597.Google Scholar
Spillane, N., Dolean, V., Hauret, P., Nataf, F., Pechstein, C. and Scheichl, R. (2014), Abstract robust coarse spaces for systems of PDEs via generalized eigenproblems in the overlaps, Numer . Math. 126, 741–770.Google Scholar
Szyld, D. B. (2006), The many proofs of an identity on the norm of oblique projections, Numer . Algorithms 42, 309–323.CrossRefGoogle Scholar
Tan, W. C. and Hoang, V. H. (2019), High dimensional finite element method for multiscale nonlinear monotone parabolic equations, J. Comput. Appl. Math. 345, 471–500.CrossRefGoogle Scholar
Tan, W. C. and Hoang, V. H. (2020), Sparse tensor product finite element method for nonlinear multiscale variational inequalities of monotone type, IMA J. Numer. Anal. 40, 1875–1907.CrossRefGoogle Scholar
Verfürth, B. (2019a), Heterogeneous multiscale method for the Maxwell equations with high contrast, ESAIM Math. Model. Numer. Anal. 53, 35–61.CrossRefGoogle Scholar
Verfürth, B. (2019b), Numerical homogenization for nonlinear strongly monotone problems. Available at arXiv:1907.01883.Google Scholar
Verfürth, R. (1996), A Review of A Posteriori Error Estimation and Adaptive Mesh-Refinement Techniques, Wiley-Teubner.Google Scholar
Wiersma, D. S., Bartolini, P., Lagendijk, A. and Righini, R. (1997), Localization of light in a disordered medium, Nature 390, 671–673.CrossRefGoogle Scholar
Xia, B. and Hoang, V. H. (2014), High dimensional finite elements for multiscale wave equations, Multiscale Model . Simul. 12, 1622–1666.Google Scholar