Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-26T18:52:49.988Z Has data issue: false hasContentIssue false
  • English
  • Français

Adaptive Mixed GMsFEM for Flows in Heterogeneous Media

Part of: Partial differential equations, boundary value problems

Published online by Cambridge University Press:  17 November 2016

Ho Yuen Chan ,
Eric Chung  and
Yalchin Efendiev
Ho Yuen Chan*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR
Eric Chung*
Affiliation:
Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR
Yalchin Efendiev*
Affiliation:
Department of Mathematics and Institute for Scientific Computation (ISC), Texas A&M University, College Station, Texas 77843-3368, USA
*
*Corresponding author. Email addresses:hychan@math.cuhk.edu.hk (H. Y. Chan), tschung@math.cuhk.edu.hk (E. Chung), efendiev@math.tamu.edu (Y. Efendiev)
*Corresponding author. Email addresses:hychan@math.cuhk.edu.hk (H. Y. Chan), tschung@math.cuhk.edu.hk (E. Chung), efendiev@math.tamu.edu (Y. Efendiev)
*Corresponding author. Email addresses:hychan@math.cuhk.edu.hk (H. Y. Chan), tschung@math.cuhk.edu.hk (E. Chung), efendiev@math.tamu.edu (Y. Efendiev)
Get access

Abstract

In this paper, we present two adaptive methods for the basis enrichment of the mixed Generalized Multiscale Finite Element Method (GMsFEM) for solving the flow problem in heterogeneous media. We develop an a-posteriori error indicator which depends on the norm of a local residual operator. Based on this indicator, we construct an offline adaptive method to increase the number of basis functions locally in coarse regions with large local residuals. We also develop an online adaptive method which iteratively enriches the function space by adding new functions computed based on the residual of the previous solution and special minimum energy snapshots. We show theoretically and numerically the convergence of the two methods. The online method is, in general, better than the offline method as the online method is able to capture distant effects (at a cost of online computations), and both methods have faster convergence than a uniform enrichment. Analysis shows that the online method should start with a certain number of initial basis functions in order to have the best performance. The numerical results confirm this and show further that with correct selection of initial basis functions, the convergence of the online method can be independent of the contrast of the medium. We consider cases with both very high and very low conducting inclusions and channels in our numerical experiments.

Keywords

Mixed multiscale finite element methodsmultiscale basisadaptivityonline basisflow in heterogeneous media

MSC classification

Secondary: 65N30: Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods 65N12: Stability and convergence of numerical methods
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 4 , November 2016 , pp. 497 - 527
Copyright
Copyright © Global-Science Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Aarnes, J. E., On the use of a mixed multiscale finite element method for greater flexibility and increased speed or improved accuracy in reservoir simulation, SIAM J. Multiscale Modeling and Simulation, 2 (2004), pp. 421–439.Google Scholar
[2] Arbogast, T., Numerical subgrid upscaling of two-phase flow in porous media, in Numerical treatment of multiphase flows in porous media, Springer, 2000.Google Scholar
[3] Arbogast, T., Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal., 42 (2004), pp. 576–598.Google Scholar
[4] Chen, Z. and Hou, T., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comp., 72 (2003), pp. 541–576.CrossRefGoogle Scholar
[5] Chu, C.-C., Graham, I. and Hou, T.-Y., A new multiscale finite element method for high-contrast elliptic interface problems, Math. Comp., 79 (2010), pp. 1915–1955.Google Scholar
[6] Chung, E. and Leung, W. T., A sub-grid structure enhanced discontinuous Galerkin method for multiscale diffusion and convection-diffusion problems, Comm. Comput. Phys., 14 (2013), pp. 370–392.Google Scholar
[7] Chung, E., Efendiev, Y., and Hou, T. Y., Adaptive multiscale model reduction with generalized multiscale finite element methods, J. Comp. Phys., 320 (2016), pp. 69–95.Google Scholar
[8] Chung, E. T. and Efendiev, Y., Reduced-contrast approximations for high-contrast multiscale flow problems, Multiscale Modeling & Simulation, 8 (2010), pp. 1128–1153.Google Scholar
[9] Chung, E. T., Efendiev, Y., and Gibson, R. Jr., An energy-conserving discontinuous multiscale finite element method for the wave equation in heterogeneous media, Advances in Adaptive Data Analysis, 3 (2011), pp. 251–268.Google Scholar
[10] Chung, E. T., Efendiev, Y., and Lee, C. S., Mixed generalized multiscale finite element methods and applications, Multiscale Model. Simul., 13 (2015), pp. 338–366.Google Scholar
[11] Chung, E. T., Efendiev, Y., and Leung, W. T., An adaptive generalized multiscale discontinuous Galerkin method (GMsDGM) for high-contrast flow problems, arXiv preprint arXiv:1409.3474.Google Scholar
[12] Chung, E. T., Efendiev, Y., and Leung, W. T., Generalized multiscale finite element methods for wave propagation in heterogeneous media, Multiscale Modeling & Simulation, 12 (2014), pp. 1691–1721.Google Scholar
[13] Chung, E. T., Efendiev, Y., and Leung, W. T., An online generalized multiscale discontinuous Galerkin method (GMsDGM) for flows in heterogeneous media, To appear in Comm. Comput. Phys.Google Scholar
[14] Chung, E. T., Efendiev, Y., and Leung, W. T., Residual-driven online generalized multiscale finite element methods, J. Comp. Phys., 302 (2015), pp. 176–190.Google Scholar
[15] Chung, E. T., Efendiev, Y., and Li, G., An adaptive GMsFEM for high-contrast flow problems, J. Comp. Phys., 273 (2014), pp. 54–76.Google Scholar
[16] Chung, E.T., Efendiev, Y., and Fu, S., Generalized multiscale finite element method for elasticity equations, International Journal on Geomathematics, 5 (2014), pp. 225–254.Google Scholar
[17] Durlofsky, L. J., Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water resources research, 27 (1991), pp. 699–708.CrossRefGoogle Scholar
[18] W. E., and Engquist, B., Heterogeneous multiscale methods, Communications in Mathematical Sciences, 1 (2003), pp. 87–132.Google Scholar
[19] Efendiev, Y., Galvis, J., and Wu, X.-H., Multiscale finite element methods for high-contrast problems using local spectral basis functions, J. Comp. Phys., 230 (2011), pp. 937–955.Google Scholar
[20] Efendiev, Y. and Hou, T.-Y., Multiscale finite element methods: theory and applications, Volume 4, Surveys and tutorials in the applied mathematical sciences, Springer, 2009.Google Scholar
[21] Efendiev, Y., Hou, T.-Y., Ginting, V., Multiscale finite element methods for nonlinear problems and their applications, Communications in Mathematical Sciences, 2 (2004), pp. 553–589.Google Scholar
[22] Gao, K., Chung, E. T., Gibson, R., Fu, S. and Efendiev, Y., A numerical homogeneization method for heterogenous, anisotropic elastic media based on multiscale theory, Geophysics, 80 (2015), pp. D385–D401.Google Scholar
[23] Gao, K., Fu, S., Gibson, R., Chung, E. T. and Efendiev, Y., Generalized multiscale finite element method (GMsFEM) for elastic wave propagation in heterogeneous, anisotropic media, J. Comput. Phys., 295 (2015), pp. 161–188.Google Scholar
[24] Ghommem, M., Presho, M., Calo, V. M. and Efendiev, Y., Mode decomposition methods for flows in high-contrast porous media. Global–local approach, J. Comp. Phys., 253 (2013), pp. 226–238.Google Scholar
[25] Gibson, R., Gao, K., Chung, E. and Efendiev, Y., Multiscale modeling of acoustic wave propagation in two-dimensional media, Geophysics, 79 (2014), pp. T61–T75.Google Scholar
[26] Hou, T. Y. and Wu, X.-H., A multiscale finite element method for elliptic problems in composite materials and porous media, J. Comp. Phys., 134 (1997), pp. 169–189.Google Scholar
[27] Jenny, P., Lee, S.-H. and Tchelepi, H., Multi-scale finite volume method for elliptic problems in subsurface flow simulation, J. Comput. Phys., 187 (2003), pp. 47–67.Google Scholar
[28] Wu, X.-H., Efendiev, Y. and Hou, T.-Y., Analysis of upscaling absolute permeability, Discrete and Continuous Dynamical Systems Series B, 2 (2002), pp. 185–204.Google Scholar