2017
DOI: 10.7494/opmath.2017.37.2.327
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The interaction between PDE and graphs in multiscale modeling

Abstract: Abstract.In this article an upscaling model is presented for complex networks with highly clustered regions exchanging/trading quantities of interest at both, microscale and macroscale level. Such an intricate system is approximated by a partitioned open map in R 2 or R 3. The behavior of the quantities is modeled as flowing in the map constructed and thus it is subject to be described using partial differential equations. We follow this approach using the Darcy Porous Media, saturated fluid flow model in mixe… Show more

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Cited by 3 publications
(5 citation statements)
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“…In particular, if θ = Const., then system (1.1) reduces to the classical Navier-Stokes system which describes the motion of incompressible viscous fluid flows and has been extensively studied by many authors; see [18][19][20][21][22][23][24] and the references therein. In addition to this, the reader is referred to [25][26][27][28][29] to find more results about the related fluid flow equations.…”
Section: Introductionmentioning
confidence: 99%
“…In particular, if θ = Const., then system (1.1) reduces to the classical Navier-Stokes system which describes the motion of incompressible viscous fluid flows and has been extensively studied by many authors; see [18][19][20][21][22][23][24] and the references therein. In addition to this, the reader is referred to [25][26][27][28][29] to find more results about the related fluid flow equations.…”
Section: Introductionmentioning
confidence: 99%
“…Remark 1. (i) Clearly X is a Hilbert space, in order to see that Y is a Hilbert space see Lemma 2 above and/or Lemma 4.4 in [11].…”
Section: The Strong Problem and Its Continuous Weak Formulationmentioning
confidence: 99%
“…All the aforementioned works, whichever the problem they may be analyzing (elasticity, heat diffusion, free flow, Darcy flow, etc), treat separately the primal and dual mixed formulations (see [8], [9], [10]). The present paper is focused on using simultaneously both fundamental versions for the treatment of multiscale problems in Darcy flow (see Problem (1)), it is therefore a primal-dual mixed formulation; in a way this article is the numerical implementation of the formulation introduced in [11] (see also [12], [13] for related formulations). The stability aspects become particularly critical when dealing with multiscale problems, as the presence of physical coefficients with different orders of magnitude adds up to the built-in complexity of the mixed variational formulations (coefficient a(·) in Problem (1)).…”
Section: Introductionmentioning
confidence: 99%
“…Different forms of (1) have been considered by several researchers, see, for example, [10][11][12][13][14][15]. Forchheimer in his work [5] established three empirical nonlinear formulas: -∇p = αv + β|v|v, -∇p = αv + β|v|v + γ |v| 2 v, and -∇p = αv + β|v| m-1 v, m ∈ (1, 2), to account for the inadequacies of Darcy law with α, β, and γ empirical constants deduced from experimental data.…”
Section: Introductionmentioning
confidence: 99%
“…The applications of nonlinear differential equations are seen in many fields (see [1][2][3][4]). In the field of fluid dynamics, the nonlinear correction to Darcy's law has been an active area of research for many years.…”
Section: Introductionmentioning
confidence: 99%