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An Online Generalized Multiscale Discontinuous Galerkin Method (GMsDGM) for Flows in Heterogeneous Media

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  07 February 2017

Eric T. Chung ,
Yalchin Efendiev  and
Wing Tat Leung
Eric T. Chung*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Hong Kong
Yalchin Efendiev*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA Numerical Porous Media SRI Center, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia
Wing Tat Leung*
Affiliation:
Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA
*
*Corresponding author.Email addresses:tschung@math.cuhk.edu.hk (E. T. Chung), efendiev@math.tamu.edu (Y. Efendiev), leungwt@math.tamu.edu (W. T. Leung)
*Corresponding author.Email addresses:tschung@math.cuhk.edu.hk (E. T. Chung), efendiev@math.tamu.edu (Y. Efendiev), leungwt@math.tamu.edu (W. T. Leung)
*Corresponding author.Email addresses:tschung@math.cuhk.edu.hk (E. T. Chung), efendiev@math.tamu.edu (Y. Efendiev), leungwt@math.tamu.edu (W. T. Leung)
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Abstract

Offline computation is an essential component in most multiscale model reduction techniques. However, there are multiscale problems in which offline procedure is insufficient to give accurate representations of solutions, due to the fact that offline computations are typically performed locally and global information is missing in these offline information. To tackle this difficulty, we develop an online local adaptivity technique for local multiscale model reduction problems. We design new online basis functions within Discontinuous Galerkin method based on local residuals and some optimally estimates. The resulting basis functions are able to capture the solution efficiently and accurately, and are added to the approximation iteratively. Moreover, we show that the iterative procedure is convergent with a rate independent of physical scales if the initial space is chosen carefully. Our analysis also gives a guideline on how to choose the initial space. We present some numerical examples to show the performance of the proposed method.

Keywords

Multiscale methoddiscontinuous Galerkin methodonline basis functionsheterogeneous media

MSC classification

Secondary: 65N12: Stability and convergence of numerical methods 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Type
Research Article
Information
Communications in Computational Physics , Volume 21 , Issue 2 , February 2017 , pp. 401 - 422
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Abdulle, A.. Discontinuous Galerkin finite element heterogeneous multiscale method for elliptic problems with multiple scales. Math. Comp., 81:687–713, 2012.Google Scholar
[2] Abdulle, A. and Bai, Y.. Adaptive reduced basis finite element heterogeneous multiscale method. Comput. Methods Appl. Mech. Engrg., 257:203–220, 2013.Google Scholar
[3] Arbogast, T.. Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems. SIAM J. Numer. Anal., 42(2):576–598 (electronic), 2004.Google Scholar
[4] Brenner, S. and Scott, L.. The Mathematical Theory of Finite Element Methods. Springer-Verlag, New York, 2007.Google Scholar
[5] Chen, S. S., Donoho, D. L., and Saunders, M. A.. Atomic decomposition by basis pursuit. SIAM Rev., 43(1):129–159, 2001. Reprinted from SIAM J. Sci. Comput. 20 (1998), no. 1, 33–61 (electronic) [MR1639094 (99h:94013)].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(272):1915–1955, 2010.CrossRefGoogle Scholar
[7] Chung, E. and Efendiev, Y.. Reduced-contrast approximations for high-contrast multiscale flow problems. Multiscale Model. Simul., 8:1128–1153, 2010.Google Scholar
[8] Chung, E., Efendiev, Y., and Gibson, R.. An energy-conserving discontinuous multiscale finite element method for the wave equation in heterogeneous media. Advances in Adaptive Data Analysis, 3:251–268, 2011.Google Scholar
[9] Chung, E., Efendiev, Y., and Leung, T.. Residual-driven online generalized multiscale finite element methods. submitted, arXiv:1501.04565.Google Scholar
[10] Chung, E., Efendiev, Y., and Leung, W. T.. Generalized multiscale finite element method for wave propagation in heterogeneous media. Multiscale Model. Simul, 12:1691–1721, 2014.Google Scholar
[11] Chung, E., Efendiev, Y., and Li, G.. An adaptive GMsFEM for high contrast flow problems. J. Comput. Phys., 273:54–76, 2014.Google Scholar
[12] Chung, E. and Leung, W. T.. A sub-grid structure enhanced discontinuous galerkin method for multiscale diffusion and convection-diffusion problems. Commun. Comput. Phys., 14:370–392, 2013.Google Scholar
[13] Dorfler, W.. A convergent adaptive algorithm for poisson's equation. SIAM J. Numer. Anal., 33:1106–1124, 1996.Google Scholar
[14] Drohmann, M., Haasdonk, B., and Ohlberger, M.. Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation. SIAM J. Sci. Comput., 34(2):A937–A969, 2012.Google Scholar
[15] Durlofsky, L.J.. Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media. Water Resour. Res., 27:699–708, 1991.Google Scholar
[16] E, W. and Engquist, B.. Heterogeneous multiscale methods. Comm. Math. Sci., 1(1):87–132, 2003.Google Scholar
[17] Efendiev, Y. and Galvis, J.. A domain decomposition preconditioner for multiscale high-contrast problems. In Huang, Y., Kornhuber, R., Widlund, O., and Xu, J., editors, Domain Decomposition Methods in Science and Engineering XIX, volume 78 of Lect. Notes in Comput. Science and Eng., pages 189–196. Springer-Verlag, 2011.Google Scholar
[18] Efendiev, Y., Galvis, J., and Hou, T.. Generalized multiscale finite element methods. Journal of Computational Physics, 251:116–135, 2013.Google Scholar
[19] Efendiev, Y., Galvis, J., Lazarov, R., Moon, M., and Sarkis, M.. Generalized multiscale finite element method. Symmetric interior penalty coupling. J. Comput. Phys., 255:1–15, 2013.Google Scholar
[20] Efendiev, Y., Galvis, J., Lazarov, R., and Willems, J.. Robust domain decomposition preconditioners for abstract symmetric positive definite bilinear forms. ESAIM Math. Model. Numer. Anal., 46(5):1175–1199, 2012.Google Scholar
[21] Efendiev, Y., Galvis, J., Li, G., and Presho, M.. Generalized multiscale finite element methods. oversampling strategies. to appear in International Journal for Multiscale Computational Engineering.Google Scholar
[22] Efendiev, Y., Galvis, J., and Wu, X.H.. Multiscale finite element methods for high-contrast problems using local spectral basis functions. Journal of Computational Physics, 230:937–955, 2011.CrossRefGoogle Scholar
[23] Efendiev, Y. and Hou, T.. Multiscale Finite Element Methods: Theory and Applications, volume 4 of Surveys and Tutorials in the Applied Mathematical Sciences. Springer, New York, 2009.Google Scholar
[24] Efendiev, Y., Hou, T., and Ginting, V.. Multiscale finite element methods for nonlinear problems and their applications. Comm. Math. Sci., 2:553–589, 2004.CrossRefGoogle Scholar
[25] Elfverson, D., Georgoulis, E. H., and Malqvist, A.. An adaptive discontinuous Galerkin multiscale method for elliptic problems. Multiscale Model. Simul., 11:747–765, 2013.CrossRefGoogle Scholar
[26] Elfverson, D., Georgoulis, E. H., Malqvist, A., and Peterseim, D.. Convergence of a discontinuous Galerkin multiscale method. SIAM J. Numer. Anal., 51:3351–3372, 2013.Google Scholar
[27] Galvis, J. and Efendiev, Y.. Domain decomposition preconditioners for multiscale flows in high contrast media. reduced dimension coarse spaces. SIAM J. Multiscale Modeling and Simulation, 8:1621–1644, 2010.Google Scholar
[28] Gao, K., Chung, E., Gibson, R., Fu, S., and Efendiev, Y.. A numerical homogeneization method for heterogenous, anisotropic elastic media based on multiscale theory. Geophysics, 80:D385–D401, 2015.Google Scholar
[29] Ghommem, M., Presho, M., Calo, V. M., and Efendiev, Y.. Mode decomposition methods for flows in high-contrast porous media. global-local approach. Journal of Computational Physics, Vol. 253., pages 226–238.Google Scholar
[30] Henning, P., Ohlberger, M., and Schweizer, B.. An adaptive multiscale finite element method. Multiscale Model. Simul., 12:1078–1107, 2014.Google Scholar
[31] Hou, T. and Wu, X.H.. A multiscale finite element method for elliptic problems in composite materials and porous media. J. Comput. Phys., 134:169–189, 1997.Google Scholar
[32] Huynh, D. B. P., Knezevic, D. J., and Patera, A. T.. A static condensation reduced basis element method: approximation and a posteriori error estimation. ESAIM Math. Model. Numer. Anal., 47(1):213–251, 2013.Google Scholar
[33] Malqvist, A. and Peterseim, D.. Localization of elliptic multiscale problems. Math. Comp., 83:2583–2603, 2014.Google Scholar
[34] Mekchay, K. and Nochetto, R. H.. Convergence of adaptive finite elementmethod for general second order elliptic PDEs. SIAM J. Numer. Anal., 43:1803–1827, 2005.Google Scholar
[35] Nguyen, N. C., Rozza, G., Huynh, D. B. P., and Patera, A. T.. Reduced basis approximation and a posteriori error estimation for parametrized parabolic PDEs: application to real-time Bayesian parameter estimation. In Large-scale inverse problems and quantification of uncertainty, Wiley Ser. Comput. Stat., pages 151–177. Wiley, Chichester, 2011.Google Scholar
[36] Ohlberger, M. and Schindler, F.. Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment. arXiv:1501.05202.Google Scholar
[37] Riviere, B.. Discontinuous Galerkin Methods For Solving Elliptic And parabolic Equations: Theory and Implementation. SIAM, 2008.Google Scholar
[38] Tonn, T., Urban, K., and Volkwein, S.. Comparison of the reduced-basis and POD a posteriori error estimators for an elliptic linear-quadratic optimal control problem. Math. Comput. Model. Dyn. Syst., 17(4):355–369, 2011.Google Scholar
[39] Wu, X.H., Efendiev, Y., and Hou, T.Y.. Analysis of upscaling absolute permeability. Discrete and Continuous Dynamical Systems, Series B., 2:158–204, 2002.Google Scholar