An averaging algorithm for solving elliptic problems with discontinuous coefficients

Zaslavsky, Mikhail

doi:10.1134/s1028335808030117

Cited by 4 publications

(2 citation statements)

References 7 publications

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“…To fix the idea, we assume that the reference fine grid operator A is obtained via second-order finite-differences or finite element discretization. In general, such methods do not require to have a grid conforming to the medium discontinuities and allow (sometimes with some loss of convergence order) to use different homogenization and error correction techniques to handle curvilinear interfaces, e.g., [31,34,25]. However, they require many of degrees of freedom per wavelength for accurate enough approximation.…”

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“…Conventional multi-scale (MS) approaches are targeted to applications where the discretization of spatial operators has to be performed over large-scale domain with multiple small-scale heterogeneities and where accurate approximation of fine-scale effects is not required. Variations of MS methods include MS finite elements, or superelements [15,16,22,3], MS finite volume [23] as well as averaging algorithms [31,34]. MS methods allow to capture small-scale effects of the composite media on medium-scale computational grid (so-called coarse grid).…”

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Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations

Druskin¹,

Mamonov²,

Zaslavsky³

2017

Multiscale Model. Simul.

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We developed a novel reduced-order multi-scale method for solving large timedomain wavefield simulation problems. Our algorithm consists of two main stages. During the first "off-line" stage the fine-grid operator (of the graph Laplacian type is partitioned on coarse cells (subdomains). Then projection-type multi-scale reduced order models (ROMs) are computed for the coarse cell operators. The off-line stage is embarrassingly parallel as ROM computations for the subdomains are independent of each other. It also does not depend on the number of simulated sources (inputs) and it is performed just once before the entire time-domain simulation. At the second "on-line" stage the time-domain simulation is performed within the obtained multi-scale ROM framework. The crucial feature of our formulation is the representation of the ROMs in terms of matrix Stieltjes continued fractions (S-fractions). The layered structure of the S-fraction introduces several hidden layers in the ROM representation, that results in the block-tridiagonal dynamic system within each coarse cell.This allows us to sparsify the obtained multi-scale subdomain operator ROMs and to reduce the communications between the adjacent subdomains which is highly beneficial for a parallel implementation of the on-line stage. Our approach suits perfectly the high performance computing architectures, however in this paper we present rather promising numerical results for a serial computing implementation only. These results include 3D acoustic and multi-phase anisotropic elastic problems.

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confidence: 99%

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confidence: 99%

Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations

Druskin¹,

Mamonov²,

Zaslavsky³

2017

Multiscale Model. Simul.

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Asymptotic Analysis of the Inflow to a Fracture in an Oil–Gas Reservoir with Bottom Water

Kanevskaya,

Kuznetsov,

Ryzhova

2024

Fluid Dyn

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The study of relative phase-permeability functions for anisotropic media

Pergament

Tomin

2012

Math Models Comput Simul

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Properties of relative phase permeability functions for different characteristic configura tions of a medium with anisotropic filtration properties are studied using the numerical analytical method. The anisotropy may be related to fracturing created, e.g., as a result of a brittle rupture. Cor relations between tensors of the phase and absolute permeabilities are analyzed on the assumption of capillary equilibrium. Examples of relative phase permeability functions are obtained for media with orthotropic and monoclinic symmetry of filtration properties. The influence of the property of con nectivity on the found functions is shown, and the dependence of the position of the principal axes of the phase permeability tensor on saturation is also investigated. Thereby, the misalignment of the ten sors of the phase and absolute permeabilities is shown. The results qualitatively agree with the experi mental data.

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An averaging algorithm for solving elliptic problems with discontinuous coefficients

Cited by 4 publications

References 7 publications

Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations

Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations

Asymptotic Analysis of the Inflow to a Fracture in an Oil–Gas Reservoir with Bottom Water

The study of relative phase-permeability functions for anisotropic media

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