Mixed Multiscale Finite Element Methods Using Limited Global Information

In this paper, we study domain decomposition preconditioners for multiscale flows in high contrast media. Our problems are motivated by porous media applications where low conductivity regions play an important role in determining flow patterns. We consider flow equations governed by elliptic equations in heterogeneous media with large contrast between high and low conductivity regions. This contrast brings an additional small scale (in addition to small spatial scales) into the problem expressed as the ratio between low and high conductivity values. Using weighted coarse projections, we show that the condition number of the preconditioned systems using domain decomposition methods is independent of the contrast. For this purpose, Poincaré inequalities for weighted norms are proved in the paper. The results are further generalized by employing extension theorems from homogenization theory. Our numerical observations confirm the theoretical results.

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“…It is common to use upscaled or multiscale models to solve flow and transport processes on the coarse grid [1,2,4,5,6,7,8,11,23].…”

Section: Introductionmentioning

confidence: 99%

Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

Galvis¹,

Efendiev²

2010

Multiscale Model. Simul.

208

234

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“…To develop multiscale methods that converge without resonance errors, some limited global information is needed in the construction of basis functions. In the papers [1,2,13,19,27], limited global information has been successfully used for developing MsFEMs that converge without resonance errors. These methods are applicable for problems without scale separation.…”

Section: Introductionmentioning

confidence: 99%

“…In the paper, we use a general framework of the mixed MsFEM with multiple global information following [2]. In [2], the global fields are computed based on fine-scale solutions.…”

Section: Introductionmentioning

confidence: 99%

Mixed multiscale finite element methods using approximate global information based on partial upscaling

2009

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The use of limited global information in multiscale simulations is needed when there is no scale separation. Previous approaches entail fine-scale simulations in the computation of the global information.The computation of the global information is expensive. In this paper, we propose the use of approximate global information based on partial upscaling. A requirement for partial homogenization is to capture long-range (non-local) effects present in the finescale solution, while homogenizing some of the smallest scales. The local information at these smallest scales is captured in the computation of basis functions. Thus, the proposed approach allows us to avoid the computations at the scales that can be homogenized. This results in coarser problems for the computation of global fields. We analyze the convergence of the proposed method. Mathematical formalism is introduced, which allows estimating the errors due to small scales that are homogenized. The proposed method is applied to simulate two-phase flows in heterogeneous porous media. Numerical results are presented for various permeability fields, including those generated using two-point correlation functions and channelized permeability fields from the SPE Comparative Project (Christie and Blunt, SPE Reserv Evalu Eng 4:308-317, 2001). We consider simple cases where one can identify the scales that can be homogenized. For more general cases, we suggest the use of upscaling on the coarse grid with the size smaller than the target coarse grid where multiscale basis functions are constructed. This intermediate coarse grid renders a partially upscaled solution that contains essential non-local information. Numerical examples demonstrate that the use of approximate global information provides better accuracy than purely local multiscale methods.

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“…We also note that the incorporation of limited global information into the multiscale simulation has been realized a very effective approach in complex dynamic system. Related study and application appeared in many recent works (Chen et al 2003;Aarnes 2004;Aarnes et al 2008). …”

Section: Notes To Coarse-scale Basis Functionmentioning

confidence: 99%

Application of multiscale finite element method in the uncertainty qualification of large-scale groundwater flow

Shi

Yang

Zeng

2011

Stoch Environ Res Risk Assess

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In this article, we discuss the application of multiscale finite element method (MsFEM) to groundwater flow in heterogeneous porous media. We investigate the ability of MsFEM in qualifying the flow uncertainty. Monte Carlo simulation is employed to implement the stochastic analysis, and MsFEM is used to avoid a full resolution to the spatial variable conductivity field. Large-scale flow with high variability is investigated by inspecting the single realization as well as the probability distribution functions of head and velocity. The numerical results show that the performance of MsFEM depends on the ratio between the correlation length and the coarse element size. An accurate prediction to the velocity requires a much lower ratio than the head. The MsFEM has different convergence rates for the head and the velocity, while the convergence rates do not deteriorate as the variance grows.

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Mixed Multiscale Finite Element Methods Using Limited Global Information

Cited by 74 publications

References 23 publications

Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

Domain Decomposition Preconditioners for Multiscale Flows in High-Contrast Media

Mixed multiscale finite element methods using approximate global information based on partial upscaling

Application of multiscale finite element method in the uncertainty qualification of large-scale groundwater flow

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