Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations

Arshad, Muhammad Nadeem; Park, Eun Jae; Shin, Dong wook

doi:10.1016/j.camwa.2017.09.031

Cited by 11 publications

(5 citation statements)

References 33 publications

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“…Singlephase Lowcontrast (KIM et al, 2007;YOTOV, 2009;ARBOGAST;XIAO, 2013;PENCHEVA et al, 2013;GANIS et al, 2014a;SHIN, 2018;PARK, 2020) Highcontrast (WHEELER; XUE; YOTOV, 2012; ARBOGAST; TAO; XIAO, 2013)…”

Section: Flow Medium Articlesunclassified

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

Rocha¹

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Section: Flow Medium Articlesunclassified

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

Rocha¹

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“…In this paper, we propose efficient DD methods to solve (1.6) combining the mortar mixed finite element method (MMFEM) [10,32,55] with non-overlapping domain decomposition [1,5,45] and the L-scheme method [41,46]. Our method first reformulates (1.6) into an interface problem by eliminating the subdomain variables.…”

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confidence: 99%

Robust linear domain decomposition schemes for reduced non-linear fracture flow models

Ahmed¹,

Fumagalli²,

Budiša³

et al. 2019

Preprint

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In this work, we consider compressible single-phase flow problems in a porous media containing a fracture. In the latter, a non-linear pressure-velocity relation is prescribed. Using a non-overlapping domain decomposition procedure, we reformulate the global problem into a non-linear interface problem. We then introduce two new algorithms that are able to efficiently handle the non-linearity and the coupling between the fracture and the matrix, both based on linearization by the so-called L-scheme. The first algorithm, named MoLDD, uses the L-scheme to resolve the non-linearity, requiring at each iteration to solve the dimensional coupling via a domain decomposition approach. The second algorithm, called ItLDD, uses a sequential approach in which the dimensional coupling is part of the linearization iterations. For both algorithms, the computations are reduced only to the fracture by pre-computing, in an offline phase, a multiscale flux basis (the linear Robin-to-Neumann co-dimensional map), that represent the flux exchange between the fracture and the matrix. We present extensive theoretical findings and in particular, the stability and the convergence of both schemes are obtained, where usergiven parameters are optimized to minimise the number of iterations. Examples on two important fracture models are computed with the library PorePy and agree with the developed theory.

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“…Both approaches were originally proposed in [27] in the case of matching subdomain grids. While the pressure-mortar mixed finite element method has been extensively studied, including multiphase and multiphysics flows in porous media [36], nonlinear elliptic [11] and parabolic [10] problems, mixed elasticity [30] and poroelasticity [29], Stokes-Darcy flows [26,39], flow in fractured porous media [5], Stokes-Biot couplings [6], multiscale mortar multipoint flux mixed finite element discretizations [41], mortar mimetic finite difference methods [12], and coupling with DG methods [25], the flux-mortar mixed finite element method has only recently received an increased attention. It has been applied in the context of fracture flows [16] and coupled Stokes-Darcy flows [14,15].…”

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confidence: 99%

Flux-Mortar Mixed Finite Element Methods on NonMatching Grids

Boon¹,

Gläser²,

Helmig³

et al. 2022

SIAM J. Numer. Anal.

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The flux-mortar mixed finite element method was recently developed in [15] for a general class of domain decomposition saddle point problems on non-matching grids. In this work we develop the method for Darcy flow using the multipoint flux approximation as the subdomain discretization. The subdomain problems involve solving positive definite cell-centered pressure systems. The normal flux on the subdomain interfaces is the mortar coupling variable, which plays the role of a Lagrange multiplier to impose weakly continuity of pressure. We present well-posedness and error analysis based on reformulating the method as a mixed finite element method with a quadrature rule. We develop a non-overlapping domain decomposition algorithm for the solution of the resulting algebraic system that reduces it to an interface problem for the flux-mortar, as well as an efficient interface preconditioner. A series of numerical experiments is presented illustrating the performance of the method on general grids, including applications to flow in complex porous media.

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Analysis of multiscale mortar mixed approximation of nonlinear elliptic equations

Cited by 11 publications

References 33 publications

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

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

Robust linear domain decomposition schemes for reduced non-linear fracture flow models

Flux-Mortar Mixed Finite Element Methods on NonMatching Grids

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