Design and implementation of a multiscale mixed method based on a nonoverlapping domain decomposition procedure

Francisco, Alexandre Santos; Ginting, Victor; Pereira, Felipe; Rigelo, Joyce C.

doi:10.1016/j.matcom.2013.04.022

Cited by 33 publications

(34 citation statements)

References 14 publications

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“…Also targeting the resolution of multiscale problems by mixed methods, we recall the works in [10,11], using the domain decomposition approach, and interelement connection by pressure traces, and in [12], using Robin interface condition. The two-scale locally conservative subgrid upscaling method analysed in [13] is very close to the MHM-H(div) method, both using local Neumann mixed solvers.…”

Section: Introductionmentioning

confidence: 99%

A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers

Durán

Devloo

Gomes

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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Multiscale Hybrid Mixed (MHM) method refers to a numerical technique targeted to approximate systems of differential equations with strongly varying solutions. For fluid flow, normal fluxes (multiplier) over macro element boundaries, and coarse piecewise constant potential approximations in each macro element are computed (upscaling). Then, small details are resolved by local problems, using fine representations inside the macro elements, setting the multiplier as Neumann boundary conditions (downscaling). In this work a variant of the method is developed, denoted by MHM-H(div), adopting mixed finite elements at the dowscaling stage, instead of continuous finite elements used in all previous publications of the method. Thus, this alternative MHM method inherits improvements typical of mixed methods, as better flux accuracy, and local mass conservation at the micro scale level inside the macro elements which are important properties for multi-phase flows in rough heterogeneous media. Different two-scale stable space settings are considered. Vector face functions are supposed to have normal components restricted to a given finite dimensional trace space defined over the macro element boundaries. In each macro element, the internal flux components, with vanishing normal traces, and the potential approximations, may be enriched in different extents: with respect to internal mesh size, internal polynomial degree, or both, the choice being determined by the problem at hands. A unified general error analysis of the MHM-H(div) method is presented for all these two-scale space scenarios. Both MHM versions are compared for 2D test problems, with smooth solutions, for convergence rates verification, and for Darcy's flow in heterogeneous media. MHM-H(div) 3D simulations are presented for a known singular Darcy's solution, using adaptive macro partitions, and for an oscillatory permeability scenario.

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Section: Introductionmentioning

confidence: 99%

A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers

Durán

Devloo

Gomes

et al. 2019

Computer Methods in Applied Mechanics and Engineering

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show abstract

“…The past years have seen remarkable advances in the multiscale methods research for modeling of flows in heterogeneous porous media. Multiscale methods have been developed in the framework of finite volume (JENNY; LEE; TCHELEPI, 2003; JENNY; LEE; TCHELEPI, 2005; WANG; HAJIBEYGI; TCHELEPI, 2016; CORTINOVIS; JENNY, 2017), finite element (HOU; WU, 1997; AARNES; HOU, 2002; EFENDIEV; GALVIS; HOU, 2013), and mixed finite element methods HOU, 2003;AARNES, 2006;ARBOGAST et al, 2007a;FRANCISCO et al, 2014;. Many applications to two-phase flows can be found, for example, in (EFENDIEV et al, 2006;KROGSTAD;LIE, 2006;GANIS et al, 2014c;LIE, 2016;GALVIS, 2016;DURÁN et al, 2020) and references therein.…”

Section: Multiscale Methodsmentioning

confidence: 99%

“…Singlephase Highcontrast (FRANCISCO et al, 2014;GUIRALDELLO et al, 2018;GUIRALDELLO et al, 2019;AKBARI et al, 2019;GUIRALDELLO et al, 2020;ALSADIG et al, 2020) Multiphase or blackoil Highcontrast (ROCHA et al, 2020b) Fractured (ROCHA et al, 2020a) Source: Research data.…”

Section: Flow Medium Articlesmentioning

confidence: 99%

“…The Multiscale Robin Coupled method is a non-overlapping multiscale domain decomposition method that generalizes the Multiscale Mixed Method (MuMM) (FRANCISCO et al, 2014), which is based on a domain decomposition method that was first introduced by Douglas et al (DOUGLAS et al, 1993). The main idea of these methods is to subdivide the domain Ω in N non-overlapping subdomains Ω i , i = 1, 2, .…”

Section: Solving the Elliptic Equation By The Mrcmmentioning

confidence: 99%

“…A continuous flux at the fine scale is necessary for applications involving the transport of species, such as pollutants (in single-phase flows) or saturation (in two-phase flows and more complex models). Solutions for this problem include post-processing fluxes using downscaling techniques (GUIRALDELLO et al, 2020;FRANCISCO et al, 2014;TCHELEPI, 2003). Some strategies of this type have already been applied to the MRCM in (GUIRALDELLO et al, 2020).…”

Section: Velocity Post-processing (Downscaling)mentioning

confidence: 99%

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Enhanced multiscale mixed methods for two-phase flows in high-contrast porous media

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Conditioning by Projection for the Sampling from Prior Gaussian Distributions

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Design and implementation of a multiscale mixed method based on a nonoverlapping domain decomposition procedure

Cited by 33 publications

References 14 publications

A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers

A multiscale hybrid method for Darcy’s problems using mixed finite element local solvers

Enhanced multiscale mixed methods for two-phase flows in high-contrast porous media

Conditioning by Projection for the Sampling from Prior Gaussian Distributions

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