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Adaptive Mixed GMsFEM for Flows in Heterogeneous Media

  • Ho Yuen Chan (a1), Eric Chung (a1) and Yalchin Efendiev (a2)
Abstract
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.

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Corresponding author
*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)
References
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[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. 421439.
[2] Arbogast T., Numerical subgrid upscaling of two-phase flow in porous media, in Numerical treatment of multiphase flows in porous media, Springer, 2000.
[3] Arbogast T., Analysis of a two-scale, locally conservative subgrid upscaling for elliptic problems, SIAM J. Numer. Anal., 42 (2004), pp. 576598.
[4] Chen Z. and Hou T., A mixed multiscale finite element method for elliptic problems with oscillating coefficients, Math. Comp., 72 (2003), pp. 541576.
[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. 19151955.
[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. 370392.
[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. 6995.
[8] Chung E. T. and Efendiev Y., Reduced-contrast approximations for high-contrast multiscale flow problems, Multiscale Modeling & Simulation, 8 (2010), pp. 11281153.
[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. 251268.
[10] Chung E. T., Efendiev Y., and Lee C. S., Mixed generalized multiscale finite element methods and applications, Multiscale Model. Simul., 13 (2015), pp. 338366.
[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.
[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. 16911721.
[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.
[14] Chung E. T., Efendiev Y., and Leung W. T., Residual-driven online generalized multiscale finite element methods, J. Comp. Phys., 302 (2015), pp. 176190.
[15] Chung E. T., Efendiev Y., and Li G., An adaptive GMsFEM for high-contrast flow problems, J. Comp. Phys., 273 (2014), pp. 5476.
[16] Chung E.T., Efendiev Y., and Fu S., Generalized multiscale finite element method for elasticity equations, International Journal on Geomathematics, 5 (2014), pp. 225254.
[17] Durlofsky L. J., Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media, Water resources research, 27 (1991), pp. 699708.
[18] W. E. and Engquist B., Heterogeneous multiscale methods, Communications in Mathematical Sciences, 1 (2003), pp. 87132.
[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. 937955.
[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.
[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. 553589.
[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. D385D401.
[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. 161188.
[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. 226238.
[25] Gibson R., Gao K., Chung E. and Efendiev Y., Multiscale modeling of acoustic wave propagation in two-dimensional media, Geophysics, 79 (2014), pp. T61T75.
[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. 169189.
[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. 4767.
[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. 185204.
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Numerical Mathematics: Theory, Methods and Applications
  • ISSN: 1004-8979
  • EISSN: 2079-7338
  • URL: /core/journals/numerical-mathematics-theory-methods-and-applications
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