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

Zhengkang He, Eric T. Chung, Jie Chen, Zhangxin Chen
2020 arXiv   pre-print
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  ...  A series of numerical examples are provided to highlight the performance of both these two adaptive methods and also validate the theoretical analysis.  ...  Recently, a generalized multiscale approximation of a mixed finite element method (MFEM) with velocity elimination has been developed in [27] for the subsurface flow problem, which also follows the GMsFEM  ... 
arXiv:2007.08934v1 fatcat:kfilpurzyvfszn53c4yntmob3u

Generalized multiscale approximation of a multipoint flux mixed finite element method for Darcy-Forchheimer model [article]

Zhengkang He, Eric T. Chung, Jie Chen, Zhangxin Chen
2020 arXiv   pre-print
The problem is solved in the framework of generalized multiscale finite element methods (GMsFEM) combined with a multipoint flux mixed finite element (MFMFE) method. appropriate mixed finite element spaces  ...  We consider the MFMFE method that utilizes the lowest order Brezzi-Douglas-Marini (BDM_1) mixed finite element spaces for the velocity and pressure approximation.  ...  [13] utilized the mixed generalized multiscale finite element method (mixed GMsFEM) to approximate the Darcy-Forchheimer model on the coarse grid.  ... 
arXiv:2007.08942v1 fatcat:opanrqhir5cn3l23vtautmokg4

Multiscale Methods for Model Order Reduction of Non Linear Multiphase Flow Problems [article]

Gurpreet Singh, Wingtat Leung, Mary F. Wheeler
2018 arXiv   pre-print
An expanded mixed finite element formulation is used to separate the spatial scales between non-linear, flow and transport problems.  ...  We attempt to present a comparison between two such model order reduction techniques, namely: (1) adaptive numerical homogenization and (2) generalized multiscale basis functions.  ...  Generalized mixed multiscale method In this subsection, we will present a numerical result of the generalized mixed multiscale method for black oil equation.  ... 
arXiv:1803.03721v1 fatcat:z2xtzt7xx5caxepbzpb6g2xywm

A comparison of multiscale methods for elliptic problems in porous media flow

Vegard Kippe, Jørg E. Aarnes, Knut-Andreas Lie
2008 Computational Geosciences  
We review and perform comparison studies for three recent multiscale methods for solving elliptic problems in porous media flow; the multiscale mixed finite-element method, the numerical subgrid upscaling  ...  We therefore choose to also compare the multiscale methods with a state-of-theart upscaling method-the adaptive local-global upscaling method, which may be viewed as a multiscale method when coupled with  ...  We are also grateful to the reviewers of this paper for numerous helpful comments and suggestions.  ... 
doi:10.1007/s10596-007-9074-6 fatcat:yseiamirmzcn7fcipv57r4bhea

Implementation of a mortar mixed finite element method using a Multiscale Flux Basis

Benjamin Ganis, Ivan Yotov
2009 Computer Methods in Applied Mechanics and Engineering  
This paper provides a new implementation of a multiscale mortar mixed finite element method for second order elliptic problems.  ...  The gain in computational efficiency increases with the number of subdomains.  ...  Our choice of mixed finite element method for the discretization is motivated by its local elementwise conservation of mass and accurate velocity approximation.  ... 
doi:10.1016/j.cma.2009.09.009 fatcat:miyz4c2tufe57g2tugnd4ooaku

Multiscale Simulations for Coupled Flow and Transport Using the Generalized Multiscale Finite Element Method

Eric Chung, Yalchin Efendiev, Wing Leung, Jun Ren
2015 Computation  
We construct a coarse grid solver based on the Generalized Multiscale Finite Element Method (GMsFEM) for a coupled system.  ...  Our numerical results show that with only a few basis functions per coarse block, we can achieve a good approximation.  ...  equations within the context of Generalized Multiscale Finite Element Method (GMsFEM).  ... 
doi:10.3390/computation3040670 fatcat:ljm5o3wgdrf5bhejpfayka7x34

Adaptive multiscale model reduction with Generalized Multiscale Finite Element Methods

Eric Chung, Yalchin Efendiev, Thomas Y. Hou
2016 Journal of Computational Physics  
We present a general adaptive multiscale model reduction framework, the Generalized Multiscale Finite Element Method.  ...  In this paper, we discuss a general multiscale model reduction framework based on multiscale finite element methods. We give a brief overview of related multiscale methods.  ...  The proposed methods take their origin in Multiscale Finite Element Methods [50, 54] and Generalized Finite Element Methods [60] .  ... 
doi:10.1016/j.jcp.2016.04.054 fatcat:b3xi4jz5ezhlvpvtqkd5wnhwx4

A Multiscale Method for Modeling Transport in Porous Media on Unstructured Corner-Point Grids

Jørg E. Aarnes, Yalchin Efendiev
2008 Journal of Algorithms & Computational Technology  
In this method the global flow is computed on a coarse grid scale, but information from a fine scale velocity field is used to improve accuracy.  ...  In this paper we propose a multiscale method for solving transport equations on a coarse grid.  ...  The elliptic part is solved on a coarse grid with a multiscale mixed finite element method [5, 3] . This method provides high resolution velocity fields at low computational cost.  ... 
doi:10.1260/174830108784646616 fatcat:3wyaogepkfhgbkiioqnk76t34e

A hybrid HDMR for mixed multiscale finite element method with application for flows in random porous media [article]

Lijian Jiang, J. David Moulton, Jia Wei
2013 arXiv   pre-print
To treat the heterogeneity, we use a mixed multiscale finite element method (MMsFEM) to simulate each of derived stochastic models.  ...  This significantly enhances the accuracy of the multiscale simulation.  ...  Mixed multiscale finite element method.  ... 
arXiv:1211.6510v2 fatcat:qlahhxu3hjggncjbiwqelod7iq

Multiscale Computations for Flow and Transport in Porous Media [chapter]

Thomas Y. Hou
2009 Series in Contemporary Applied Mathematics  
This paper reviews some of the recent advances in developing systematic multiscale methods with particular emphasis on multiscale finite element methods with applications to flow and transport in heterogeneous  ...  This manuscript is not intended to be a general survey paper on this topic. The discussion is limited by the scope of the lectures and expertise of the author.  ...  Mixed finite element methods with limited global information One can carry out the analysis of mixed multiscale finite element method with limited global information.  ... 
doi:10.1142/9789814273268_0003 fatcat:c2otu3os5vd43neqiopwbpfkqy

Locally conservative, stabilized finite element methods for variably saturated flow

C.E. Kees, M.W. Farthing, C.N. Dawson
2008 Computer Methods in Applied Mechanics and Engineering  
Here, we consider conforming finite element discretizations based on a multiscale formulation along with recently developed, local postprocessing schemes.  ...  The resulting approach maintains the basic flexibility and appeal of traditional finite element methods, while controlling nonphysical oscillations and producing element-wise massconservative velocity  ...  As a point of reference, we also considered a locally conservative nonconforming finite element approximation that coincides with a mixed hybrid finite element approximations in many cases.  ... 
doi:10.1016/j.cma.2008.06.005 fatcat:r2p7kyyz3zg4zpnpq3cqu6kdta

Multiscale Computations for Flow and Transport in Heterogeneous Media [chapter]

Yalchin Efendiev, Thomas Yizhao Hou
2008 Lecture notes in mathematics  
Extra effort is made in developing a multiscale computational method that can be potentially used for practical multiscale for problems with a large range of nonseparable scales.  ...  This homogenization theory provides the critical guideline for designing effective multiscale methods. In part 2, I will review some recent developments of multiscale finite element (volume) methods.  ...  Mixed finite element methods with limited global information One can carry out the analysis of mixed multiscale finite element method with limited global information.  ... 
doi:10.1007/978-3-540-79574-2_4 fatcat:wcihalffjvfobppsdjxjrlywva

Expanded Mixed Multiscale Finite Element Methods and Their Applications for Flows in Porous Media

L. Jiang, D. Copeland, J. D. Moulton
2012 Multiscale Modeling & simulation  
We develop a family of expanded mixed Multiscale Finite Element Methods (MsFEMs) and their hybridizations for second-order elliptic equations.  ...  This formulation expands the standard mixed Multiscale Finite Element formulation in the sense that four unknowns (hybrid formulation) are solved simultaneously: pressure, gradient of pressure, velocity  ...  Chen and Hou developed a local multiscale basis equation for velocity and combined it with a mixed finite element formulation to propose a mixed MsFEM [14] .  ... 
doi:10.1137/11083143x fatcat:y4na4xuipvf2rhzuin563nvmba

A multiscale multilevel mimetic (M3) method for well-driven flows in porous media

Konstantin Lipnikov, J. David Moulton, Daniil Svyatskiy
2010 Procedia Computer Science  
The multiscale multilevel mimetic (M 3 ) method was designed in [13] for the accurate modeling of two-phase flows in highly heterogeneous porous media on general polygonal meshes.  ...  A similar elimination procedure is often used in mixed-hybrid finite element methods.  ...  The multiscale multilevel mimetic (M 3 ) method [13] uses the synergy of two ideas: exact algebraic coarsening [10] and approximate (but accurate) flux coarsening, to generate a multilevel hierarchy  ... 
doi:10.1016/j.procs.2010.04.083 fatcat:jbseksm44rehtfkragksduuyha

Page 5004 of Mathematical Reviews Vol. , Issue 2001G [page]

2001 Mathematical Reviews  
Basic problems of adaptation of finite-element approximations in solving multi-dimensional problems of mathematical physics are discussed.  ...  The velocity block can be approximated by a domain decomposition method, e.g., of wire basket type, which is constructed from a local solver for each face of the elements, and a coarse solver related to  ... 
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