Adaptive generalized multiscale approximation of a mixed finite element method with velocity elimination

In this paper, we propose offline and online adaptive enrichment algorithms for the generalized multiscale approximation of a mixed finite element method with velocity elimination to solve the subsurface flow problem in high-contrast and heterogeneous porous media. We give the theoretical analysis for the convergence of these two adaptive methods, which shows that sufficient initial basis functions (belong to the offline space) leads to a faster convergence rate. A series of numerical examples are provided to highlight the performance of both these two adaptive methods and also validate the theoretical analysis. Both offline and online adaptive methods are effective that can reduce the relative error substantially. In addition, the online adaptive method generally performs better than the offline adaptive method as online basis functions contain important global information such as distant effects that cannot be captured by offline basis functions. The numerical results also show that with a suitable initial multiscale space that includes all offline basis functions corresponding to relative smaller eigenvalues of local spectral decompositions in the offline stage, the convergence rate of the online enrichment is independent of the permeability contrast.

[1]  Yalchin Efendiev,et al.  Generalized Multiscale Finite Element Methods. Oversampling Strategies , 2013, 1304.4888.

[2]  Eric T. Chung,et al.  Generalized multiscale approximation of mixed finite elements with velocity elimination for subsurface flow , 2020, J. Comput. Phys..

[3]  Yalchin Efendiev,et al.  Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods , 2016, J. Comput. Phys..

[4]  T. F. Russell,et al.  Finite element and finite difference methods for continuous flows in porous media. , 1800 .

[5]  Thomas Y. Hou,et al.  A Multiscale Finite Element Method for Elliptic Problems in Composite Materials and Porous Media , 1997 .

[6]  Eric T. Chung,et al.  A numerical homogenization method for heterogeneous, anisotropic elastic media based on multiscale theory , 2015 .

[7]  Yalchin Efendiev,et al.  Generalized multiscale finite element method. Symmetric interior penalty coupling , 2013, J. Comput. Phys..

[8]  Yalchin Efendiev,et al.  An Adaptive Generalized Multiscale Discontinuous Galerkin Method for High-Contrast Flow Problems , 2018, Multiscale Model. Simul..

[9]  H. Tchelepi,et al.  Multi-scale finite-volume method for elliptic problems in subsurface flow simulation , 2003 .

[10]  H. Owhadi,et al.  Metric‐based upscaling , 2007 .

[11]  Thomas Y. Hou,et al.  Convergence of a Nonconforming Multiscale Finite Element Method , 2000, SIAM J. Numer. Anal..

[12]  Gurpreet Singh,et al.  Multiscale methods for model order reduction of non-linear multiphase flow problems , 2018, Computational Geosciences.

[13]  Yalchin Efendiev,et al.  Generalized Multiscale Finite Element Methods for Wave Propagation in Heterogeneous Media , 2013, Multiscale Model. Simul..

[14]  Victor M. Calo,et al.  Randomized Oversampling for Generalized Multiscale Finite Element Methods , 2014, Multiscale Model. Simul..

[15]  Mary F. Wheeler,et al.  A Multipoint Flux Mixed Finite Element Method , 2006, SIAM J. Numer. Anal..

[16]  Yalchin Efendiev,et al.  Multiscale finite element methods for high-contrast problems using local spectral basis functions , 2011, J. Comput. Phys..

[17]  T. Hou,et al.  Multiscale Finite Element Methods for Nonlinear Problems and Their Applications , 2004 .

[18]  Yalchin Efendiev,et al.  Adaptive mixed GMsFEM for flows in heterogeneous media , 2015 .

[19]  Mehdi Ghommem,et al.  Mode decomposition methods for flows in high-contrast porous media. Global-local approach , 2013, J. Comput. Phys..

[20]  Yalchin Efendiev,et al.  An Energy-Conserving Discontinuous Multiscale Finite Element Method for the wave equation in Heterogeneous Media , 2011, Adv. Data Sci. Adapt. Anal..

[21]  T. Hou,et al.  Analysis of upscaling absolute permeability , 2002 .

[22]  Yerlan Amanbek,et al.  Adaptive numerical homogenization for upscaling single phase flow and transport , 2019, J. Comput. Phys..

[23]  Zhangxin Chen,et al.  Generalized multiscale approximation of a mixed finite element method with velocity elimination for Darcy flow in fractured porous media , 2021 .

[24]  Hamdi A. Tchelepi,et al.  Well Modeling in the Multiscale Finite Volume Method for Subsurface Flow Simulation , 2006, Multiscale Model. Simul..

[25]  L. Durlofsky.  Numerical calculation of equivalent grid block permeability tensors for heterogeneous porous media , 1991 .

[26]  Wing Tat Leung,et al.  A Sub-Grid Structure Enhanced Discontinuous Galerkin Method for Multiscale Diffusion and Convection-Diffusion Problems , 2013 .

[27]  Yalchin Efendiev,et al.  Generalized multiscale finite element methods (GMsFEM) , 2013, J. Comput. Phys..

[28]  Yanbo Li,et al.  Adaptive generalized multiscale finite element methods for H(curl)-elliptic problems with heterogeneous coefficients , 2018, J. Comput. Appl. Math..

[29]  M. Larson,et al.  Adaptive variational multiscale methods based on a posteriori error estimation: Energy norm estimates for elliptic problems , 2007 .

[30]  Patrick Jenny,et al.  The Multiscale Finite Volume Method: A flexible tool to model physically complex flow in porous media , 2006 .

[31]  Yalchin Efendiev,et al.  Mixed Generalized Multiscale Finite Element Methods and Applications , 2014, Multiscale Model. Simul..

[32]  Yerlan Amanbek,et al.  Error indicator for the incompressible Darcy flow problems using Enhanced Velocity Mixed Finite Element Method , 2019, ArXiv.

[33]  Yalchin Efendiev,et al.  An online generalized multiscale discontinuous Galerkin method (GMsDGM) for flows in heterogeneous media , 2015, 1504.04417.

[34]  Patrick Jenny,et al.  Iterative multiscale finite-volume method , 2008, J. Comput. Phys..

[35]  Yalchin Efendiev,et al.  An adaptive GMsFEM for high-contrast flow problems , 2013, J. Comput. Phys..

[36]  Eric T. Chung,et al.  Residual driven online mortar mixed finite element methods and applications , 2018, J. Comput. Appl. Math..

[37]  Yalchin Efendiev,et al.  Reduced-Contrast Approximations for High-Contrast Multiscale Flow Problems , 2010, Multiscale Model. Simul..

[38]  Jiangyong Hou,et al.  Numerical comparison of robustness of some reduction methods in rough grids , 2014 .

[39]  Yalchin Efendiev,et al.  Accurate multiscale finite element methods for two-phase flow simulations , 2006, J. Comput. Phys..

[40]  Yalchin Efendiev,et al.  Multiscale Finite Element Methods: Theory and Applications , 2009 .

[41]  Yalchin Efendiev,et al.  Residual-driven online generalized multiscale finite element methods , 2015, J. Comput. Phys..

[42]  Ivan G. Graham,et al.  A new multiscale finite element method for high-contrast elliptic interface problems , 2010, Math. Comput..

[43]  Ragnar Winther,et al.  Robust convergence of multi point flux approximation on rough grids , 2006, Numerische Mathematik.

[44]  Ivan Lunati,et al.  An Operator Formulation of the Multiscale Finite-Volume Method with Correction Function , 2009, Multiscale Model. Simul..

[45]  Jie Chen,et al.  Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model , 2020, ArXiv.