An analysis of the NLMC upscaling method for high contrast problems

Zhao, Lina; Chung, Eric T.

doi:10.1016/j.cam.2019.112480

Cited by 15 publications

(10 citation statements)

References 31 publications

(35 reference statements)

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“…Lemma 1 can be proved using a similar technique in Lemma 3.2 [17]. For brevity of this article, we omit the proof.…”

Section: Convergence Analysismentioning

confidence: 98%

Space-time Non-local multi-continua upscaling for parabolic equations with moving channelized media

Hu,

Leung,

Chung

et al. 2021

Preprint

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In this paper, we consider a parabolic problem with time-dependent heterogeneous coefficients. Many applied problems have coupled space and time heterogeneities. Their homogenization or upscaling requires cell problems that are formulated in space-time representative volumes for problems with scale separation. In problems without scale separation, local problems include multiple macroscopic variables and oversampled local problems, where these macroscopic parameters are computed. These approaches, called Non-local multi-continua, are proposed for problems with complex spatial heterogeneities in a number of previous papers. In this paper, we extend this approach for spacetime heterogeneities, by identifying macroscopic parameters in space-time regions. Our proposed method space-time Non-local multi-continua (space-time NLMC) is an efficient numerical solver to deal with time-dependent heterogeneous coefficients. It provides a flexible and systematic way to construct multiscale basis functions to approximate the solution. These multiscale basis functions are constructed by solving a local energy minimization problems in the oversampled space-time regions such that these multiscale basis functions decay exponentially outside the oversampled domain. Unlike the classical time-stepping methods combined with full-discretization technique, our space-time NLMC efficiently constructs the multiscale basis functions in a space-time domain and can provide a computational savings compared to space-only approaches as we discuss in the paper. We present two numerical experiments, which show that the proposed approach can provide a good accuracy.

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“…Lemma 1 can be proved using a similar technique in Lemma 3.2 [17]. For brevity of this article, we omit the proof.…”

Section: Convergence Analysismentioning

confidence: 98%

Space-time Non-local multi-continua upscaling for parabolic equations with moving channelized media

Hu,

Leung,

Chung

et al. 2021

Preprint

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“…Moreover, one can define the continua by using properties of the heterogeneous media. In this case, the auxiliary basis functions are piecewise constant functions defined with respect to a partition of the coarse cell K i , such as the medium coefficients have a bounded contrast in each subregion [47].…”

Section: Global Couplingmentioning

confidence: 99%

Space-Time Nonlinear Upscaling Framework Using Nonlocal Multicontinuum Approach

Leung

Chung

Efendiev

et al. 2019

Int J Mult Comp Eng

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In this paper, we develop a space-time upscaling framework that can be used for many challenging porous media applications without scale separation and high contrast. Our main focus is on nonlinear differential equations with multiscale coefficients. The framework is built on nonlinear nonlocal multicontinuum upscaling concept [16] and significantly extends the results in the proceeding paper [17].Our approach starts with a coarse space-time partition and identifies test functions for each partition, which play a role of multi-continua. The test functions are defined via optimization and play a crucial role in nonlinear upscaling. In the second stage, we solve nonlinear local problems in oversampled regions with some constraints defined via test functions. These local solutions define a nonlinear map from macroscopic variables determined with the help of test functions to the fine-grid fields. This map can be thought as a downscaled map from macroscopic variables to the fine-grid solution. In the final stage, we seek macroscopic variables in the entire domain such that the downscaled field solves the global problem in a weak sense defined using the test functions. We present an analysis of our approach for an example nonlinear problem.Our unified framework plays an important role in designing various upscaled methods. Because local problems are directly related to the fine-grid problems, it simplifies the process of finding local solutions with appropriate constraints [16]. Using machine learning (ML), we identify the complex map from macroscopic variables to fine-grid solution. We present numerical results for several porous media applications, including two-phase flow and transport.

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“…In more rigorous approaches related to porous media [4,27,3], the continua are assumed to be fracture and matrix regions. In our earlier works [14,13,30], we define the continua via local spectral decompositions and show that the resulting approach converges independent of scales and contrast if representative volumes are chosen to be coarse blocks. In this work, we use similar ideas (as in [14,13,30]) for problems with scale separation and formulate cell problems and formally derive multicontinuum equations.…”

Section: Introductionmentioning

confidence: 99%

Multicontinuum homogenization and its relation to nonlocal multicontinuum theories

Efendiev¹,

Leung²

2022

Preprint

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In this paper, we present a general derivation of multicontinuum equations and discuss cell problems. We present constraint cell problem formulations in a representative volume element and oversampling techniques that allow reducing boundary effects. We discuss different choices of constraints for cell problems. We present numerical results that show how oversampling reduces boundary effects. Finally, we discuss the relation of the proposed methods to our previously developed methods, Nonlocal Multicontinuum Approaches.

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An analysis of the NLMC upscaling method for high contrast problems

Cited by 15 publications

References 31 publications

Space-time Non-local multi-continua upscaling for parabolic equations with moving channelized media

Space-time Non-local multi-continua upscaling for parabolic equations with moving channelized media

Space-Time Nonlinear Upscaling Framework Using Nonlocal Multicontinuum Approach

Multicontinuum homogenization and its relation to nonlocal multicontinuum theories

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