Skip to content
Licensed Unlicensed Requires Authentication Published by De Gruyter June 11, 2016

Robust Numerical Upscaling of Elliptic Multiscale Problems at High Contrast

  • Daniel Peterseim EMAIL logo and Robert Scheichl

Abstract

We present a new approach to the numerical upscaling for elliptic problems with rough diffusion coefficient at high contrast. It is based on the localizable orthogonal decomposition of H1 into the image and the kernel of some novel stable quasi-interpolation operators with local L2-approximation properties, independent of the contrast. We identify a set of sufficient assumptions on these quasi-interpolation operators that guarantee in principle optimal convergence without pre-asymptotic effects for high-contrast coefficients. We then give an example of a suitable operator and establish the assumptions for a particular class of high-contrast coefficients. So far this is not possible without any pre-asymptotic effects, but the optimal convergence is independent of the contrast and the asymptotic range is largely improved over other discretization schemes. The new framework is sufficiently flexible to allow also for other choices of quasi-interpolation operators and the potential for fully robust numerical upscaling at high contrast.

MSC 2010: 65N30; 65N25; 65N15

Funding statement: Supported by the Sino-German Science Center (grant id 1228) on the occasion of the Chinese-German Workshop on Computational and Applied Mathematics in Augsburg 2015.

Acknowledgements

We thank Clemens Pechstein for suggesting the alternative, projective quasi-interpolation operator and providing us with the basic ideas for its analysis.

References

[1] Babuška I. and Lipton R., The penetration function and its application to microscale problems, Multiscale Model. Simul. 9 (2011), no. 1, 373–406. 10.1007/s10543-008-0182-zSearch in Google Scholar

[2] Berlyand L. and Owhadi H., Flux norm approach to finite dimensional homogenization approximations with non-separated scales and high contrast, Arch. Ration. Mech. Anal. 198 (2010), 677–721. 10.1007/s00205-010-0302-1Search in Google Scholar

[3] Brown D. and Peterseim D., A multiscale method for porous microstructures, preprint 2014, http://arxiv.org/abs/1411.1944. 10.1137/140995210Search in Google Scholar

[4] Carstensen C., Quasi-interpolation and a posteriori error analysis in finite element methods, M2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1187–1202. 10.1051/m2an:1999140Search in Google Scholar

[5] Carstensen C. and Verfürth R., Edge residuals dominate a posteriori error estimates for low order finite element methods, SIAM J. Numer. Anal. 36 (1999), no. 5, 1571–1587. 10.1137/S003614299732334XSearch in Google Scholar

[6] Chu C.-C., Graham I. G. and Hou T.-Y., A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp. 79 (2010), no. 272, 1915–1955. 10.1090/S0025-5718-2010-02372-5Search in Google Scholar

[7] Dryja M., Sarkis M. V. and Widlund O. B., Multilevel Schwarz methods for elliptic problems with discontinuous coefficients in three dimensions, Numer. Math. 72 (1996), no. 3, 313–348. 10.1007/s002110050172Search in Google Scholar

[8] E W. and Engquist B., The heterogeneous multiscale methods, Commun. Math. Sci. 1 (2003), no. 1, 87–132. 10.4310/CMS.2003.v1.n1.a8Search in Google Scholar

[9] Efendiev Y., Galvis J. and Hou T. Y., Generalized multiscale finite element methods, J. Comput. Phys. 251 (2013), 116–135. 10.1016/j.jcp.2013.04.045Search in Google Scholar

[10] Elfverson D., Georgoulis E. H., Målqvist A. and Peterseim D., Convergence of a discontinuous Galerkin multiscale method, SIAM J. Numer. Anal. 51 (2013), no. 6, 3351–3372. 10.1137/120900113Search in Google Scholar

[11] Gallistl D. and Peterseim D., Stable multiscale Petrov–Galerkin finite element method for high frequency acoustic scattering, Comput. Methods Appl. Mech. Engrg. 295 (2015), 1–17. 10.1016/j.cma.2015.06.017Search in Google Scholar

[12] Henning P. and Målqvist A., Localized orthogonal decomposition techniques for boundary value problems, SIAM J. Sci. Comput. 36 (2014), no. 4, A1609–A1634. 10.1137/130933198Search in Google Scholar

[13] Henning P., Målqvist A. and Peterseim D., A localized orthogonal decomposition method for semi-linear elliptic problems, ESAIM Math. Model. Numer. Anal. 48 (2014), no. 5, 1331–1349. 10.1051/m2an/2013141Search in Google Scholar

[14] Henning P., Morgenstern P. and Peterseim D., Multiscale partition of unity, Meshfree Methods for Partial Differential Equations VII, Lect. Notes Comput. Sci. Eng. 100, Springer, Cham (2015), 185–204. 10.1007/978-3-319-06898-5_10Search in Google Scholar

[15] Henning P. and Peterseim D., Oversampling for the multiscale finite element method, Multiscale Model. Simul. 11 (2013), no. 4, 1149–1175. 10.1137/120900332Search in Google Scholar

[16] Hou T. Y. and Wu X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comput. Phys. 134 (1997), no. 1, 169–189. 10.1006/jcph.1997.5682Search in Google Scholar

[17] Hughes T. J. R., Feijóo G. R., Mazzei L. and Quincy J.-B., The variational multiscale method—a paradigm for computational mechanics, Comput. Methods Appl. Mech. Engrg. 166 (1998), 3–24. 10.1016/S0045-7825(98)00079-6Search in Google Scholar

[18] Hughes T. J. R. and Sangalli G., 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. 10.1137/050645646Search in Google Scholar

[19] Kornhuber R. and Yserentant H., Numerical homogenization of elliptic multiscale problems by subspace decomposition, preprint 2015. 10.1137/15M1028510Search in Google Scholar

[20] Målqvist A. and Peterseim D., Computation of eigenvalues by numerical upscaling, Numer. Math. 130 (2014), no. 2, 337–361. 10.1007/s00211-014-0665-6Search in Google Scholar

[21] Målqvist A. and Peterseim D., Localization of elliptic multiscale problems, Math. Comp. 83 (2014), no. 290, 2583–2603. 10.1090/S0025-5718-2014-02868-8Search in Google Scholar

[22] Owhadi H. and Zhang L., Localized bases for finite-dimensional homogenization approximations with nonseparated scales and high contrast, Multiscale Model. Simul. 9 (2011), no. 4, 1373–1398. 10.1137/100813968Search in Google Scholar

[23] Owhadi H., Zhang L. and Berlyand L., Polyharmonic homogenization, rough polyharmonic splines and sparse super-localization, ESAIM Math. Model. Numer. Anal. 48 (2013), no. 2, 517–552. 10.1051/m2an/2013118Search in Google Scholar

[24] Pechstein C. and Scheichl R., Weighted Poincaré inequalities, IMA J. Numer. Anal. 33 (2012), no. 2, 652–686. 10.1093/imanum/drs017Search in Google Scholar

[25] Peterseim D., Composite finite elements for elliptic interface problems, Math. Comp. 83 (2014), no. 290, 2657–2674. 10.1090/S0025-5718-2014-02815-9Search in Google Scholar

[26] Peterseim D., Eliminating the pollution effect in Helmholtz problems by local subscale correction, preprint 2014, http://arxiv.org/abs/1411.7512. 10.1090/mcom/3156Search in Google Scholar

[27] Peterseim D., Variational multiscale stabilization and the exponential decay of fine-scale correctors, preprint 2015, http://arxiv.org/abs/1505.07611. 10.1007/978-3-319-41640-3_11Search in Google Scholar

[28] Scheichl R., Vassilevski P. S. and Zikatanov L. T., Weak approximation properties of elliptic projections with functional constraints, Multiscale Model. Simul. 9 (2011), no. 4, 1677–1699. 10.1137/110821639Search in Google Scholar

[29] Scheichl R., Vassilevski P. S. and Zikatanov L. T., Mutilevel methods for elliptic problems with highly varying coefficients on non-aligned coarse grids, SIAM J. Numer. Anal. 50 (2012), 1675–1694. 10.1137/100805248Search in Google Scholar

Received: 2016-1-24
Revised: 2016-5-14
Accepted: 2016-5-15
Published Online: 2016-6-11
Published in Print: 2016-10-1

© 2016 by De Gruyter

Downloaded on 17.5.2024 from https://www.degruyter.com/document/doi/10.1515/cmam-2016-0022/html
Scroll to top button