Open Access
Issue |
ESAIM: M2AN
Volume 55, 2021
Regular articles published in advance of the transition of the journal to Subscribe to Open (S2O). Free supplement sponsored by the Fonds National pour la Science Ouverte
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Page(s) | S149 - S185 | |
DOI | https://doi.org/10.1051/m2an/2020024 | |
Published online | 26 February 2021 |
- A. Abdulle, D. Arjmand and E. Paganoni, Exponential decay of the resonance error in numerical homogenization via parabolic and elliptic cell problems. C. R. Math. Acad. Sci. Paris 357 (2019) 545–551. [Google Scholar]
- M.A. Akcoglu and U. Krengel, Ergodic theorems for superadditive processes. J. Reine Angew. Math. 323 (1981) 53–67. [CrossRef] [MathSciNet] [Google Scholar]
- Y. Almog, Averaging of dilute random media: a rigorous proof of the Clausius-Mossotti formula. Arch. Ration. Mech. Anal. 207 (2013) 785–812. [Google Scholar]
- Y. Almog, The Clausius-Mossotti formula in a dilute random medium with fixed volume fraction. Multiscale Model. Simul. 12 (2014) 1777–1799. [Google Scholar]
- Y. Almog, The Clausius-Mossotti formula for dilute random media of perfectly conducting inclusions. SIAM J. Math. Anal. 49 (2017) 2885–2919. [Google Scholar]
- A. Anantharaman and C. Le Bris, A numerical approach related to defect-type theories for some weakly random problems in homogenization. Multiscale Model. Simul. 9 (2011) 513–544. [Google Scholar]
- A. Anantharaman and C. Le Bris, Elements of mathematical foundations for numerical approaches for weakly random homogenization problems. Commun. Comput. Phys. 11 (2012) 1103–1143. [Google Scholar]
- S. Armstrong and P. Dario, Elliptic regularity and quantitative homogenization on percolation clusters. Commun. Pure Appl. Math. 71 (2018) 1717–1849. [Google Scholar]
- S.N. Armstrong and J.-C. Mourrat, Lipschitz regularity for elliptic equations with random coefficients. Arch. Ration. Mech. Anal. 219 (2016) 255–348. [Google Scholar]
- S.N. Armstrong and C.K. Smart, Quantitative stochastic homogenization of convex integral functionals. Ann. Sci. Éc. Norm. Supér. (4) 49 (2016) 423–481. [Google Scholar]
- S. Armstrong, T. Kuusi and J.-C. Mourrat, Mesoscopic higher regularity and subadditivity in elliptic homogenization. Commun. Math. Phys. 347 (2016) 315–361. [Google Scholar]
- S. Armstrong, T. Kuusi and J.-C. Mourrat, The additive structure of elliptic homogenization. Invent. Math. 208 (2017) 999–1154. [Google Scholar]
- S. Armstrong, T. Kuusi and J.-C. Mourrat, Quantitative Stochastic Homogenization and Large-Scale Regularity. In: Vol. 352 of Grundlehren der mathematischen Wissenschaften, Springer Nature (2019). [CrossRef] [Google Scholar]
- B.K. Bergen and F. Hülsemann, Hierarchical hybrid grids: data structures and core algorithms for multigrid. Numer. Linear Algebra App. 11 (2004) 279–291. [Google Scholar]
- B. Bergen, F. Hülsemann and U. Rüde, Is 1.7 × 1010 unknowns the largest finite element system that can be solved today? In: SC’05: Proceedings of the 2005 ACM/IEEE Conference on Supercomputing. IEEE (2005). [Google Scholar]
- L. Berlyand and V. Mityushev, Generalized Clausius-Mossotti formula for random composite with circular fibers. J. Stat. Phys. 102 (2001) 115–145. [Google Scholar]
- J. Bey, Tetrahedral grid refinement. Computing 55 (1995) 355–378. [CrossRef] [MathSciNet] [Google Scholar]
- X. Blanc and C. Le Bris, Improving on computation of homogenized coefficients in the periodic and quasi-periodic settings. Netw. Heterogen. Media 5 (2010) 1–29. [Google Scholar]
- X. Blanc, C. Le Bris and, F. Legoll, Some variance reduction methods for numerical stochastic homogenization. Philos. Trans. R. Soc. A 374 (2016) 15. [Google Scholar]
- S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, 3rd edition. In: Vol. 15 of Texts in Applied Mathematics. Springer, New York (2008). [CrossRef] [Google Scholar]
- E. Cancès, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, An embedded corrector problem for homogenization. Part II: algorithms and discretization. J. Comput. Phys. 407 (2020) 109254,26. [Google Scholar]
- E. Cancès, V. Ehrlacher, F. Legoll, B. Stamm and S. Xiang, An embedded corrector problem for homogenization. Part I: theory. Preprint arXiv:1807.05131 (2018). [Google Scholar]
- P. Dario, Optimal corrector estimates on percolation clusters. Preprint arXiv:1805.00902 (2020). [Google Scholar]
- M. Duerinckx and A. Gloria, Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas. Arch. Ration. Mech. Anal. 220 (2016) 297–361. [Google Scholar]
- Y. Efendiev and T.Y. Hou, Multiscale finite element methods. In: Vol. 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York (2009). [Google Scholar]
- A.-C. Egloffe, A. Gloria, J.-C. Mourrat and T.N. Nguyen, Random walk in random environment, corrector equation and homogenized coefficients: from theory to numerics, back and forth. IMA J. Numer. Anal. 35 (2015) 499–545. [Google Scholar]
- J. Fischer, The choice of representative volumes in the approximation of effective properties of random materials. Arch. Ration. Mech. Anal. 234 (2019) 635–726. [Google Scholar]
- A. Gholami, D. Malhotra, H. Sundar and G. Biros, FFT, FMM, or multigrid? A comparative study of state-of-the-art Poisson solvers for uniform and nonuniform grids in the unit cube. SIAM J. Sci. Comput. 38 (2016) C280–C306. [Google Scholar]
- A. Gloria, Numerical approximation of effective coefficients in stochastic homogenization of discrete elliptic equations. ESAIM: M2AN 46 (2012) 1–38. [EDP Sciences] [Google Scholar]
- A. Gloria and Z. Habibi, Reduction in the resonance error in numerical homogenization II: correctors and extrapolation. Found. Comput. Math. 16 (2016) 217–296. [Google Scholar]
- A. Gloria and J.-C. Mourrat, Spectral measure and approximation of homogenized coefficients. Probab. Theory Relat. Fields 154 (2012) 287–326. [Google Scholar]
- A. Gloria and F. Otto, An optimal variance estimate in stochastic homogenization of discrete elliptic equations. Ann. Probab. 39 (2011) 779–856. [Google Scholar]
- A. Gloria and F. Otto, An optimal error estimate in stochastic homogenization of discrete elliptic equations. Ann. Appl. Probab. 22 (2012) 1–28. [Google Scholar]
- A. Gloria and F. Otto, Quantitative results on the corrector equation in stochastic homogenization. J. Eur. Math. Soc. (JEMS) 19 (2017) 3489–3548. [Google Scholar]
- A. Gloria and F. Otto, The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations. Preprint arXiv:1510.08290 (2016). [Google Scholar]
- A. Gloria, S. Neukamm and F. Otto, Quantification of ergodicity in stochastic homogenization: optimal bounds via spectral gap on Glauber dynamics. Invent. Math. 199 (2015) 455–515. [Google Scholar]
- A. Gloria, S. Neukamm and F. Otto, A regularity theory for random elliptic operators. Preprint arXiv:1409.2678 (2019). [Google Scholar]
- T. Gradl and U. Rüde, High performance multigrid on current large scale parallel computers. In: 9th Workshop on Parallel Systems and Algorithms (2008). [Google Scholar]
- A. Hannukainen, J.-C. Mourrat and H. Stoppels, Homogenization.jl tutorial. Available from: https://haampie.github.io/Homogenization.jl/dev/ (2020). [Google Scholar]
- T.Y. Hou and X.-H. Wu, A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys. 134 (1997) 169–189. [Google Scholar]
- T.Y. Hou, X.-H. Wu and Z. Cai, Convergence of a multiscale finite element method for elliptic problems with rapidly oscillating coefficients. Math. Comput. 68 (1999) 913–943. [Google Scholar]
- V. Khoromskaia, B.N. Khoromskij and F. Otto, Numerical study in stochastic homogenization for elliptic PDEs: convergence rate in the size of representative volume elements. Preprint arXiv:1903.12227 (2019). [Google Scholar]
- S.M. Kozlov, Geometric aspects of averaging. Uspekhi Mat. Nauk 44 (1989) 79–120. [Google Scholar]
- C. Le Bris and F. Legoll, Examples of computational approaches for elliptic, possibly multiscale PDEs with random inputs. J. Comput. Phys. 328 (2017) 455–473. [Google Scholar]
- C. Le Bris, F. Legoll and W. Minvielle, Special quasirandom structures: a selection approach for stochastic homogenization. Monte Carlo Methods App. 22 (2016) 25–54. [Google Scholar]
- D. Marahrens and F. Otto, Annealed estimates on the Green function. Probab. Theory Relat. Fields 163 (2015) 527–573. [Google Scholar]
- J.C. Maxwell, Medium in which small spheres are uniformly disseminated, 3rd edition. In: A Treatise on Electricity and Magnetism, part II, chapter IX. Clarendon Press (1891) 314. [Google Scholar]
- N.G. Meyers, An Lp-estimate for the gradient of solutions of second order elliptic divergence equations. Ann. Scuola Norm. Sup. Pisa 17 (1963) 189–206. [MathSciNet] [Google Scholar]
- J.-C. Mourrat, Variance decay for functionals of the environment viewed by the particle. Ann. Inst. Henri Poincaré Probab. Stat. 47 (2011) 294–327. [Google Scholar]
- J.-C. Mourrat, First-order expansion of homogenized coefficients under Bernoulli perturbations. J. Math. Pures Appl. 103 (2015) 68–101. [Google Scholar]
- J.-C. Mourrat, Efficient methods for the estimation of homogenized coefficients. Found. Comput. Math. 19 (2019) 435–483. [Google Scholar]
- J.-C. Mourrat, An informal introduction to quantitative stochastic homogenization. J. Math. Phys. 60 (2019) 11. [Google Scholar]
- A. Naddaf and T. Spencer, On homogenization and scaling limit of some gradient perturbations of a massless free field. Commun. Math. Phys. 183 (1997) 55–84. [Google Scholar]
- A. Naddaf and T. Spencer, Estimates on the variance of some homogenization problems (1998). [Google Scholar]
- G.C. Papanicolaou, Diffusion in random media. In: Vol. 1 of Surveys in Applied Mathematics. Plenum, New York (1995) 205–253. [Google Scholar]
- L.C. Piccinini and S. Spagnolo, On the Hölder continuity of solutions of second order elliptic equations in two variables. Ann. Scuola Norm. Sup. Pisa 26 (1972) 391–402. [Google Scholar]
- J.W. Strutt, 3d Baron Rayleigh, On the influence of obstacles arranged in rectangular order upon the properties of a medium. Philos. Mag. 34 (1892) 481–502. [CrossRef] [Google Scholar]
- S.-H. Wei, L. Ferreira, J.E. Bernard and A. Zunger, Electronic properties of random alloys: special quasirandom structures. Phys. Rev. B 42 (1990) 9622. [Google Scholar]
- X. Yue and E. Weinan, The local microscale problem in the multiscale modeling of strongly heterogeneous media: effects of boundary conditions and cell size. J. Comput. Phys. 222 (2007) 556–572. [Google Scholar]
- A. Zunger, S.-H. Wei, L. Ferreira and J.E. Bernard, Special quasirandom structures. Phys. Rev. Lett. 65 (1990) 353. [CrossRef] [PubMed] [Google Scholar]
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