Mixed and Hybrid Finite Element Methods

Fortin, M.

doi:10.1007/978-1-4612-3172-1

Cited by 3,878 publications

(83 citation statements)

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“…[31] The numerical model of the flow in the entire network, resolved at the fracture scale, is based on the Mixed Finite Element method, mainly for two reasons: it ensures both local and global mass conservation and it provides an accurate velocity field, which can be used in subsequent transport simulations. We implemented the so-called RT0 scheme [Brezzi and Fortin, 1991;Raviart and Thomas, 1977]. The networks considered have a very specific geometry: it is a 3D intricate structure of 2D domains.…”

Section: Flow Modeling and Simulation In The 3d Fracture Networkmentioning

confidence: 99%

Influence of fracture scale heterogeneity on the flow properties of three‐dimensional discrete fracture networks (DFN)

2012

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[1] While permeability scaling of fractured media has been so far studied independently at the fracture-and network-scales, we propose a numerical analysis of the combined effect of fracture-scale heterogeneities and the network-scale topology. The analysis is based on 2Á10 6 discrete fracture network (DFNs) simulations performed with highly robust numerical methods. Fracture local apertures are distributed according to a truncated Gaussian law, and exhibit self-affine spatial correlations up to a cutoff scale L c . Network structures range widely over sparse and dense systems of short, long or widely distributed fracture sizes and display a large variety of fracture interconnections, flow bottlenecks and dead-ends. At the fracture scale, accounting for aperture heterogeneities leads to a reduction of the equivalent fracture transmissivity of up to a factor of 6 as compared to the parallel plate of identical mean aperture. At the network scale, a significant coupling is observed in most cases between flow heterogeneities at the fracture and at the network scale. The upscaling from the fracture to the network scale modifies the impact of fracture roughness on the measured permeability. This change can be quantified by the measure a 2, which is analogous to the more classical power-averaging exponent used with heterogeneous porous media, and whose magnitude results from the competition of two effects: (i) the permeability is enhanced by the highly transmissive zones within the fractures that can bridge fracture intersections within a fracture plane; (ii) it is reduced by the closed and low transmissive areas that break up connectivity and flow paths.

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Section: Flow Modeling and Simulation In The 3d Fracture Networkmentioning

confidence: 99%

Influence of fracture scale heterogeneity on the flow properties of three‐dimensional discrete fracture networks (DFN)

2012

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“…The 1 0 finite element method is particularly controversial [1]. Despite being damned by theoreticians after the discovery of "weakly singular" modes [5,6], 1 0 is widely used in practice.…”

Section: Q P mentioning

confidence: 99%

“…Applying the analysis of Brezzi and Fortin [1] or Malkus [7] the inf-sup stability of the system (4) (or, equivalently, (5)) is determined by the square root of the smallest non zero eigenvalue of (6) that is…”

Section: Finite Element Approximationmentioning

confidence: 99%

“…It is universally recognized that discretization schemes for Stokes and Navier-Stokes equations are subject to an inf-sup or div-stability condition [1]. The stability requirement is manifested in practical computations by the predominance of staggered grid finite volume discretizations, and the existence of unnatural velocity-pressure finite element combinations.…”

Section: Introductionmentioning

confidence: 99%

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A Finite Element Approximation of the Stokes Equations

Alami¹,

Elakkad²,

El-Mekaoui³

et al. 2011

WJM

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In this work, a numerical solution of the incompressible Stokes equations is proposed. The method suggested is based on an algorithm of discretization by the unstable of Q 1 -P 0 velocity/pressure finite element approximation. It is shown that the inf-sup stability constant is O(h) in two dimensions and O( ) in three dimensions. The basic tool in the analysis is the method of modified equations which is applied to finite difference representations of the underlying finite element equations. In order to evaluate the performance of the method, the numerical results are compared with some previously published works or with others coming from commercial code like Adina system.

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“…The traditional approach of deriving MFE coefficients assumes that the resulting bilinear forms, derived initially for the entire domain by a variational analysis, can be applied piecewise to the partitioned domain (e.g., [4,5]). The bilinear forms, in turn, provide that the test function is applied over the entire element T i , as opposed to a control volume associated with the principal face in the CVMFE method.…”

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confidence: 99%

A control volume mixed finite element formulation for tetrahedral elements

Naff

2015

Numerical Meth Engineering

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SummaryThe control‐volume mixed finite element method is formulated for and applied to a computational domain consisting of a tetrahedral partitioning to solve the steady groundwater flow equations. Test functions consistent with piecewise constant and piecewise linear pressure distributions are used in the formulation. Comparisons are made with a standard mixed finite element formulation using lowest‐order Raviart Thomas basis functions. Results suggest that the control‐volume based formulation is a viable alternative to the standard formulation. Copyright © 2015 John Wiley & Sons, Ltd.

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Mixed and Hybrid Finite Element Methods

Cited by 3,878 publications

References 0 publications

Influence of fracture scale heterogeneity on the flow properties of three‐dimensional discrete fracture networks (DFN)

Influence of fracture scale heterogeneity on the flow properties of three‐dimensional discrete fracture networks (DFN)

A Finite Element Approximation of the Stokes Equations

A control volume mixed finite element formulation for tetrahedral elements

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