Vivette Girault's research while affiliated with French National Centre for Scientific Research and other places
What is this page?
This page lists the scientific contributions of an author, who either does not have a ResearchGate profile, or has not yet added these contributions to their profile.
It was automatically created by ResearchGate to create a record of this author's body of work. We create such pages to advance our goal of creating and maintaining the most comprehensive scientific repository possible. In doing so, we process publicly available (personal) data relating to the author as a member of the scientific community.
If you're a ResearchGate member, you can follow this page to keep up with this author's work.
If you are this author, and you don't want us to display this page anymore, please let us know.
It was automatically created by ResearchGate to create a record of this author's body of work. We create such pages to advance our goal of creating and maintaining the most comprehensive scientific repository possible. In doing so, we process publicly available (personal) data relating to the author as a member of the scientific community.
If you're a ResearchGate member, you can follow this page to keep up with this author's work.
If you are this author, and you don't want us to display this page anymore, please let us know.
Publications (170)
Flow coupled with geomechanics problems has gathered increased research interest due to its resemblance to engineering applications, such as unconventional reservoir development, by incorporating multiple physics. Computations for the system of such a multiphysics model is often costly. In this paper, we introduce a posteriori error estimators to g...
We first derive convergence and a priori stability, and next reliability and efficiency of a posteriori error indicators for a Biot poroelastic model coupled with an elastic model in R3, solved by a continuous Galerkin scheme (CG) for the displacement and a mixed finite element scheme for the flow. The coupled system is decoupled by a fixed stress...
We focus on the ill posed data completion problem and its finite element approximation, when recast via the variational duplication Kohn-Vogelius artifice and the condensation Steklov-Poincaré operators. We try to understand the useful hidden features of both exact and discrete problems. When discretized with finite elements of degree one, the disc...
Coupled subsurface fluid flow and geomechanics is receiving growing research interests for applications in geothermal energy, unconventional oil and gas recovery and geological CO2 sequestration. A key model characterizing these processes is the Biot system. In this paper, we present optimal L2 error estimates for the Biot system. The flow equation...
We prove an asymptotic relationship between the grade-two fluid model and a class of models for non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwell models. This confirms an earlier observation of Tanner. We provide a new interpretation of the temporal instability of the grade-two fluid model for nega...
We derive optimal reliability and efficiency of a posteriori error estimates for the steady Stokes problem, with a nonhomogeneous Dirichlet boundary condition, solved by a stable enriched Galerkin scheme (EG) of order one on triangular or quadrilateral meshes in ℝ ² , and tetrahedral or hexahedral meshes in ℝ ³ .
Electroless plating in microfluidic channels is a novel technology at the micrometer scale. As the microchannel depth varies with the flow of the chemicals, care must be taken for the channel not to run dry. Owing to the deposited chemical species the physical domain of the flow changes with time, leading to a free boundary problem. As the motion o...
This paper presents the numerical solution of immiscible two-phase flows in porous media, obtained by a first-order finite element method equipped with mass-lumping and flux upwinding. The unknowns are the physical phase pressure and phase saturation. Our numerical experiments confirm that the method converges optimally for manufactured solutions....
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We...
This paper presents the numerical solution of immiscible two-phase flows in porous media, obtained by a first-order finite element method equipped with mass-lumping and flux up-winding. The unknowns are the physical phase pressure and phase saturation. Our numerical experiments confirm that the method converges optimally for manufactured solutions....
We consider the mathematical analysis and numerical approximation of a system of nonlinear partial differential equations that arises in models that have relevance to steady isochoric flows of colloidal suspensions. The symmetric velocity gradient is assumed to be a monotone nonlinear function of the deviatoric part of the Cauchy stress tensor. We...
A finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. The discrete saturation satisfies a maximum principle. Stability of the scheme and existence of a so...
Convergence of a finite element method with mass-lumping and flux upwinding is formulated for solving the immiscible two-phase flow problem in porous media. The method approximates directly the wetting phase pressure and saturation, which are the primary unknowns. Well-posedness is obtained in [7]. Theoretical convergence is proved via a compactnes...
Modelling the interactions between mechanical deformations and fluid flow in a porous media leads to the poromechanics [1]. It involves two coupled conservation equations obtained from the mechanical equilibrium and from the fluid mass conservation. A common way to numerically solve a poroelastic model, which is the simplest form of poromechanics,...
We analyze the Biot system solved with a fixed-stress split, Enriched Galerkin (EG) discretization for the flow equation, and Galerkin for the mechanics equation. Residual-based a posteriori error estimates are established with both lower and upper bounds. These theoretical results are confirmed by numerical experiments performed with the Mandel’s...
Coupled fluid flow and geomechanics is receiving growing research interests for applications in un-conventional oil and gas recovery and geological CO 2 sequestration. In this work, we first demonstrated the advantage of using the Enriched Galerkin (EG) method for flow discretization in unconventional scenarios where the permeability of fractures/f...
In this paper, we design and study a fully coupled numerical scheme for the poroelasticity problem modeled through Biot’s equations. The classical way to numerically solve this system is to use a finite element method for the mechanical equilibrium equation and a finite volume method for the fluid mass conservation equation. However, to capture spe...
We construct a finite element approximation of a strain-limiting elastic model on a bounded open domain in $\mathbb{R}^d$, $d \in \{2,3\}$. The sequence of finite element approximations is shown to exhibit strong convergence to the unique weak solution of the model. A rate of convergence for the sequence of finite element approximations is shown pr...
This work derives a posteriori error estimates, in two and three dimensions, for the heat equation coupled with Darcy's law by a nonlinear viscosity depending on the temperature. We introduce two variational formulations and discretize them by finite element methods. We prove optimal a posteriori errors with two types of computable error indicators...
The classical way to numerically solve a poroelastic model is to use one discretisation method for each conservation equation, usually with a finite element method for the mechanical part and a finite volume method for the fluid part. However, to capture specific properties of underground media such as heterogeneities, discontinuities and faults, m...
We consider a poro-elastic region embedded into an elastic non-porous region. The elastic displacement equations are discretized by a continuous Galerkin scheme, while the flow equations for the pressure in the poro-elastic medium are discretized by either a continuous Galerkin scheme or a mixed scheme. Since the overall system of equations is very...
We analyse and discretize a mixed formulation for a linearized lubrication fracture model in a poro-elastic medium. The displacement of the medium is expressed in primary variables while the flows in the medium and fracture are written in mixed form, with an additional unknown for the pressure in the fracture. The fracture is treated as a non-plana...
Modelling the interactions between mechanical deformations and fluid flow in a porous media leads to the well known Biot system. This system involves two coupled equations obtained from the mechanical equilibrium and from the fluid mass conservation. The classical way to numerically solve this system is to use one discretisation method for each con...
![Error curve for different ν(T)\documentclass[12pt]{minimal}...](publication/312487221/figure/fig4/AS:960034934628352@1605901573374/Error-curve-for-different-nTdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
In this article, we study theoretically and numerically the heat equation coupled with Darcy's law by a nonlinear viscosity depending on the temperature. We establish existence and uniqueness of the exact solution by using a Galerkin method. We propose and analyze two numerical schemes based on finite element methods. An optimal a priori error esti...
We construct a finite element approximation of a strain-limiting elastic model on a bounded open domain in $\mathbb{R}^d$, $d \in \{2,3\}$. The sequence of finite element approximations is shown to exhibit strong convergence to the unique weak solution of the model. Assuming that the material parameters featuring in the model are Lipschitz-continuo...
![The boundary ΓC\documentclass[12pt]{minimal} \usepackage{amsmath}...](profile/Faker-Ben-Belgacem/publication/308675460/figure/fig2/AS:960034745892865@1605901528486/The-boundary-GCdocumentclass12ptminimal-usepackageamsmath-usepackagewasysym_Q320.jpg)
![Polygonal incomplete boundary ΓI\documentclass[12pt]{minimal}...](profile/Faker-Ben-Belgacem/publication/308675460/figure/fig3/AS:960034745901089@1605901528597/Polygonal-incomplete-boundary-GIdocumentclass12ptminimal-usepackageamsmath_Q320.jpg)
The variational finite element solution of Cauchy's problem, expressed in the Steklov-Poincaré framework and regularized by the Lavrentiev method, has been introduced and computationally assessed in [Inverse Problems in Science and Engineering, 18, 1063–1086 (2011)]. The present work concentrates on the numerical analysis of the semi-discrete probl...
We propose a stable element for the divergence operator that approximates the velocity by continuous linear polynomials plus piecewise constants and the pressure by piecewise constants. A uniform inf–sup condition is obtained for conforming meshes in two or three dimensions. The resulting method belongs to the class of enriched Galerkin methods, an...
We review an argument of Renardy proving existence and regularity for a subset of a class of models of non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwellian models. We suggest an effective method for solving these models, including a variational formulation suitable for finite element computation. ©...
We modify an argument of Renardy proving existence and regularity for a subset of a class of models of non-Newtonian fluids suggested by Oldroyd, including the upper-convected and lower-convected Maxwellian models. We suggest an effective method for solving these models, which can provide a variational formulation suitable for finite element comput...
A numerical method is proposed and analyzed for the coupled time-dependent Navier-Stokes equations and Darcy equations. Existence and uniqueness of the solution is obtained under a small data condition. A priori error estimates are derived. Numerical examples confirm the theoretical convergence rates.
Strong convergence of the numerical solution to a weak solution is proved for a nonlinear coupled flow and transport problem arising in porous media. The method combines a mixed finite element method for the pressure and velocity with an interior penalty discontinuous Galerkin method in space for the concentration. Using functional tools specific t...
We consider an iterative scheme for solving a coupled geomechanics and flow problem in a fractured poroelastic medium. The fractures are treated as possibly non-planar interfaces. Our iterative scheme is an adaptation due to the presence of fractures of a classical fixed stress-splitting scheme. We prove that the iterative scheme is a contraction i...
In this chapter, we present the essential ideas of the mathematical analysis of the equations modeling the flow of grade-two fluids.
This short chapter collects most mathematical notions, definitions, and results that will be used in the following chapters. Nearly all results are recalled without proof, or are briefly established. Additional results with proofs will be found in Chapter 7.
In this chapter, the reader will be introduced to a variety of non-Newtonian phenomena exhibited by real fluids, namely stress relaxation, nonlinear creep, shear-thinning and shear-thickening, thixotropy, development of normal stress differences in simple shear flows, yield, etc.
The aim of this section is to discuss the mathematical properties of the governing equations of some non-Newtonian fluids introduced in Chapter 2, Section 2. 4, namely, the Reiner-Rivlin fluid and in particular, the Bingham fluid.
There is no space here for a detailed study of grade-three fluids. Consequently, in this chapter we briefly present a few salient results on the mathematical analysis of the equations modeling the flow of grade-three fluids. As is the case for the grade-two model, the results are more favorable in two dimensions, but for the sake of brevity, we res...
We propose and analyze a simplified fluid-stru cture coupled model for flows with compliant walls. As in [F. Nobile and C. Vergara, SIAM J. Sci. Comput., 30(2008), pp. 731-763], the wall reaction to the fluid is modeled by a small displacement viscoelastic shell where the tangential stress components and displacements are neglected. We show that wi...
A convergence analysis to the weak solution is derived for interior penalty discontinuous Galerkin methods applied to the heat equation in two and three dimensions under general mixed boundary conditions. Strong convergence is established in the DG norm, as well as in the
We analyze and discretize a mixed formulation for a linearized lubrication fracture model in a poro-elastic medium. The displacement of the medium is expressed in primary variables while the flows in the medium and fracture are written in mixed form, with an additional unknown for the pressure in the fracture. The fracture is treated as a non-plana...
This text is the first of its kind to bring together both the thermomechanics and mathematical analysis of Reiner-Rivlin fluids and fluids of grades 2 and 3 in a single book. Each part of the book can be considered as being self-contained. The first part of the book is devoted to a description of the mechanics, thermodynamics, and stability of flow...
The inf-sup constant for the divergence, or LBB constant, is explicitly known for only few domains. For other domains, upper and lower estimates are known. If more precise values are required, one can try to compute a numerical approximation. This involves, in general, approximation of the domain and then the computation of a discrete LBB constant...
This last chapter is devoted to the proofs of auxiliary results, with particular emphasis on properties of simple linear transport equations that lie at the core of many complex fluids.
In this work, we introduce a discrete specific inf-sup condition to estimate the Lp norm, 1 < p < +∞, of the pressure in a number of fluid flows. It applies to projection-based stabilized finite element discretizations of incompressible flows, typically when the velocity field has a low regularity. We derive two versions of this inf-sup condition:...
We prove error estimates in the maximum norm, namely in \(W^{1,\infty }(\Omega )^3\times L^\infty (\Omega )\), for the Stokes and Navier–Stokes equations in convex, three-dimensional domains \(\Omega \) with simplicial boundaries. We modify the weighted \(L^2\) estimates for regularized Green functions used earlier by us, which impose restrictions...
We present a non-planar fracture model in a poro-elastic medium. The medium in which the fracture is embedded is governed by the standard Biot equations of linear poro-elasticity and the flow of the fluid within the fracture is governed by the lubrication equation. We establish existence and uniqueness of the linearized coupled system under weak as...
We present a fracture model in a poro-elastic medium. The model describes the fracture as a curve or surface according to the dimension, the width of the crack being included into the equation of flow in the fracture. The discretization uses mixed finite elements for the fluid and continuous finite elements for the porous medium's displacement. The...
The coupled Stokes and Darcy equations are approximated by a strongly conservative finite element method. The discrete spaces are the divergence-conforming velocity space with matching pressure space such as the Raviart-Thomas spaces. This work proves optimal error estimate of the velocity in the L 2 norm in the domain and on the interface. Lipschi...
We investigate mortar multiscale numerical methods for coupled Stokes and Darcy flows with the Beavers-Joseph-Saffman interface condition. The domain is decomposed into a series of subdomains (coarse grid) of either Stokes or Darcy type. The subdomains are discretized by appropriate Stokes or Darcy finite elements. The solution is resolved locally...
We analyze a simplified coupled model for aortic blood flow. The vessel wall reaction to the fluid is modeled by the Surface Pressure Model which assumes that the normal stress on the fluid is proportional to the displacement of the structure. This leads to a unique boundary problem for Navier-Stokes equations, where at the wall the velocity is nor...
In dimension 2, the Horgan-Payne angle serves to construct a lower bound for the inf-sup constant of the divergence arising in the so-called LBB condition. This lower bound is equivalent to an upper bound for the Friedrichs constant. Explicit upper bounds for the latter constant can be found using a polar parametrization of the boundary. Revisiting...
In this paper, we introduce a low-cost, high-order stabilized method for the numerical solution of incompressible flow problems.
This is a particular type of projection-stabilized method where each targeted operator, such as the pressure gradient or the
convection, is stabilized by least-squares terms added to the Galerkin formulation. The main met...
A weak solution of the coupling of time-dependent incompressible Navier-Stokes equations with Darcy equations is defined. The interface conditions include the Beavers-Joseph-Saffman condition. Existence and uniqueness of the weak solution are obtained by a constructive approach. The analysis is valid for weak regularity interfaces.
In this chapter, we present the theoretical analysis of coupled incompressible Navier–Stokes (or Stokes) flows and Darcy flows with the Beavers–Joseph–Saffman interface condition. We discuss alternative interface and porous media models. We review some finite element methods used by several authors in this coupling and present numerical experiments...
This chapter discusses the numerical methods for grade-two fluid models—finite-element discretizations and algorithms. It also discusses homogeneous Dirichlet boundary conditions. It presents discretization of the steady-state problem and the discretization of the time-dependent problem. The heuristic least-squares scheme and gradient algorithm for...
We derive an a posteriori error estimate for an abstract saddle-point problem when a penalty term is added to stabilize the variational formulation, the aim being to optimize the choice of the penalty parameter. As an application, we consider a discretization of the Stokes problem obtained by combining the penalty technique and the finite element m...
We couple a time-dependent poroelastic model in a region with an elastic model in adjacent regions. We discretize each model independently on non-matching grids and we realize a domain decomposition on the interface between the regions by introducing DG jumps and mortars. The unknowns are condensed on the interface, so that at each time step, the c...
A linear fully discrete mixed scheme, using Cº finite elements in space and a semi-implicit Euler scheme in time, is considered for solving a penalized nematic liquid crystal model (of the Ginzburg-Landau type). We prove: 1) unconditional stability and convergence towards weak solutions, and 2) first-order optimal error estimates for regular soluti...
We consider the flow of a viscous incompressible fluid through a rigid homogeneous porous medium. The permeability of the medium depends on the pressure, so that the model is nonlinear. We propose a finite element discretization of this problem and, in the case where the dependence on pressure is bounded from above and below, we prove its convergen...
We consider a steady transport system of equations in a bounded Lipschitz domain of RdRd, 2⩽d⩽42⩽d⩽4, with a divergence-free transport velocity in H1H1, tangential on the boundary. By means of two regularizations, first with a viscous penalty term and next with a Yosida approximation, we prove that an LpLp data, 2⩽p<∞2⩽p<∞, yields a solution in LpL...
We prove uniqueness of the solution of a time-dependent transport equation with a divergence-free driving velocity that is L 1 in time and H 1 in space, in a Lipschitz domain of ℝ d , tangential on the boundary. The proof is done by regularization with a special mollifier.
We study a second-order two-grid scheme fully discrete in time and space for solving the Navier–Stokes equations. The two-grid strategy consists in discretizing, in the first step, the fully non-linear problem, in space on a coarse grid with mesh-size H and time step Δt and, in the second step, in discretizing the linearized problem around the velo...
In this article, we present a domain decomposition method for solving a linear elasticity system. Data are transmitted by jumps, as in a discontinuous Galerkin method, and mortars are introduced at the interfaces to dissociate the computation between neighboring subdomains. A decoupling algorithm condenses the unknowns on the interface. The matrix...
In this work, we couple the incompressible steady Navier-Stokes equations with the Darcy equations, by means of the Beaver-Joseph-Saffman's condition on the interface. Under suitable smallness conditions on the data, we prove existence of a weak solution as well as some a priori estimates. We establish local uniqueness when the data satisfy additio...
The system of unsteady Darcy's equations considered here models the time-dependent flow of an incompressible fluid such as water in a rigid porous medium. We propose a discretization of this problem that relies on a backward Euler's scheme for the time variable and finite elements for the space variables. We prove a priori error estimates that just...
We solve a steady Darcy–Forchheimer flow in a bounded region by means of piecewise constant velocities and nonconforming piecewise
\mathbbP1{\mathbb{P}_1} pressures. For the computation, we solve the nonlinearity by an alternating-directions algorithm and we decouple the computation
of the velocity from that of the pressure by a gradient algorithm...
In this article, we describe some simple and commonly used discontinuous Galerkin methods for elliptic, Stokes and convection-diffusion
problems. We illustrate these methods by numerical experiments.
Discontinuous Galerkin (DG) and mixed nite element (MFE) methods are two popular methods that possess local mass conservation. In this paper we investigate DG-DG and DG-MFE domain decomposition couplings using mortar nite elements to impose weak continuity of uxes and pressures on the interface. The subdomain grids need not match and the mortar gri...
We study the equations of motion of two immiscible fluids with comparable densities, but very different viscosities in a two-dimensional horizontal pipe. This is applied to the lubricated transportation of heavy crude oil. First, we write the problem in variational form and next we derive an energy balance for this model.
This paper is devoted to the analysis of finite element approximations of convection-diffusion equations with data in L1. We discretize the convection operator by the PSI (Positive Streamwise Implicit) scheme, and the diffusion operator by the
standard Galerkin method, using conforming P1 finite elements. We give the main idea in the proof of conve...
This article is devoted to the numerical analysis of a fictitious domain method for the Stokes problem, where the boundary
condition is enforced weakly by means of a multiplier defined in a portion of the domain. In practice, this is applied for
example to the sedimentation of many particles in a fluid. It is found that the multiplier is divergence...
Les éléments finis de Crouzeix–Raviart engendrent un espace de fonctions discontinues, affines sur chaque élément d'une triangulation du domaine. Le but de cette Note est de prouver un résultat de trace pour cet espace. On présente une application au traitement de conditions aux limites pour la discrétisation des équations de Darcy. Pour citer cet...
We prove the existence of a weak solution of a time-dependent grade-two fluid model in a plane Lipschitz domain and uniqueness of the solution in a convex polygon. The method of proof is constructive and can be adapted to the numerical analysis of finite-element schemes for solving this problem numerically.
In this paper we consider, in dimension d≥ 2, the standard $$\mathbb{P}_{1}$$ finite elements approximation of the second order linear elliptic equation in divergence form with coefficients in L
∞(Ω) which generalizes Laplace’s equation. We assume that the family of triangulations is regular and that it satisfies an
hypothesis close to the classica...
We discretize in space the equations obtained at each time step when discretizing in time a Navier-Stokes system modelling the two-dimensional flow in a horizontal pipe of two immiscible fluids with comparable densities, but very different viscosities. At each time step the system reduces to a generalized Stokes problem with nonstandard conditions...
In this paper, an improved inf–sup condition is derived for a class of discontinuous Galerkin methods for solving the steady-state incompressible Stokes and Navier–Stokes equations. The computational domain is subdivided into subdomains with non-matching meshes at the interfaces. Optimal error estimates are obtained. Numerical experiments including...
We consider the formulation of the Stokes problem in a multiply connected two-dimensional domain where the unknowns are the stream-function and the vorticity. We derive its equivalence with a finite system of several variational problems. This leads to the construction of a finite element discretization of this problem. The analysis and a numerical...
Nous complétons ici les résultats d’isomorphismes de l’opérateur de Laplace dans des espaces de Sobolev avec poids et nous donnons quelques applications. Parmi celles-ci, nous obtenons des inégalités semblables à celle de Calderon-Zygmund et en particulier des propriétés de continuité des transformées de Riesz dans des espaces avec poids. Nous donn...
In this paper we solve the time-dependent incompressible Navier-Stokes equations by splitting the non-linearity and incompressibility, and using discontinuous or continuous finite element methods in space. We prove optimal error estimates for the velocity and suboptimal estimates for the pressure. We present some numerical experiments.
In this Note, we study a two-grid scheme fully discrete in time and space for solving the Navier-Stokes system. In the first step, the fully non-linear problem is discretized in space on a coarse grid with mesh-size H and time step k. In the second step, the problem is discretized in space on a fine grid with mesh-size h and the same time step, and...
We consider a system with three unknowns in a two-dimensional bounded domain which models the flow of a grade-two non-Newtonian fluid. We propose to compute an approximation of the solution of this problem in two steps: addition of a regularization term, finite element discretization of the regularized problem. We prove optimal a priori and a poste...
We prove stability of the finite element Stokes projection in the product space W1,∞(Ω)×L∞(Ω), i.e., the maximum norm of the discrete velocity gradient and discrete pressure are bounded by the sum of the corresponding exact counterparts, independently of the mesh-size. The proof relies on weighted L2 estimates for regularized Green's functions asso...
A family of discontinuous Galerkin nite element methods,is for- mulated,and analyzed for Stokes and Navier-Stokes problems. An inf-sup condition is established as well as optimal energy estimates for the velocity and L, estimates for the pressure. In addition, it is shown that the method can treat a,nite number,of nonoverlapping domains with nonmat...
In this paper, we discuss the convergence of a domain decomposition method for the solution of linear parabolic equations in their mixed formulations. The subdomain meshes need not be quasi-uniform; they are composed of triangles or quadrilaterals that do not match at interfaces. For the ease of computation, this lack of continuity is compensated b...
We prove stability of the finite element Stokes projection in the product space W1,∞(Ω)×L∞(Ω). The proof relies on weighted L2 estimates for regularized Green's functions associated with the Stokes problem and on a weighted inf–sup condition. The domain is a polygon or a polyhedron with a Lipschitz-continuous boundary, satisfying suitable sufficien...
We consider the Navier–Stokes equations, discretized by a penalization method and finite elements. The aim of this Note is to prove a posteriori error estimates which allow for an optimal choice of the penalty parameter, specially for adaptive meshes. To cite this article: C. Bernardi et al., C. R. Acad. Sci. Paris, Ser. I 336 (2003).
This article discusses the use of stability as a guide to understanding model equations for uid ow. In particular, the end goal is to evaluate a model of uid ow which incorporates dispersion. Such models do not have a linear stress-strain law, and hence extend models of Newtonian uids. The model studied has been proposed in widely disparate context...
We construct an operator that preserves the discrete divergence and has the same quasi-local approximation properties as
a regularizing interpolant; this is very useful when discretizing nonlinear incompressible fluid models. For low-degree finite
elements, such operators have an explicit expression, from which local approximation properties can be...
Citations
... Finite element methods using Lavrientiev regularization based on this type of formulation was analysed in [20] for the elliptic Cauchy problem. It was shown in [21] that in the absence of regularization a naive choice of finite element spaces results in non-uniqueness of discrete solution. Even after regularization a nested optimization seems inevitable since both g and γ have to be determined. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/742810366005256-1554111200822_Q64/Faker-Ben-Belgacem.jpg)
![[object Object]](https://i1.rgstatic.net/ii/profile.image/564132459495424-1511511067765_Q64/Faten-Jelassi.jpg)
... It follows that the continuous model needs to be regularized before it can be useful. Finite element methods using Lavrientiev regularization based on this type of formulation was analysed in [20] for the elliptic Cauchy problem. It was shown in [21] that in the absence of regularization a naive choice of finite element spaces results in non-uniqueness of discrete solution. ...
... In the following numerical tests, we observed that the pressure has the secondorder accuracy in L 2 norm, see Tables 6.1-6.3. We may resort to recent work [53] to prove this result in a rigorous theory. This issue is an interesting topic which involve some Aubin-Nitsche arguments, but another long story. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/629229884559360-1527031503463_Q64/Xueying-Lu-5.jpg)
... Note that, in absence of bubbles, the proposed model reduces to the usual single phase model (i.e. neglecting the existence of gas) which is compatible with previous studies on electroless process such as [46]. ...
... None of the popular models available in the non-Newtonian fluid mechanics literature come close to fitting such data as the derivative of the stress-shear rate curve changes sign more than once. Recently, [61] studied mathematical issues concerning the equations governing the flow of a fluid described by (43). They established the existence of weak solutions and on making additional assumptions were able to establish uniqueness of those solutions. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/463747635519489-1487577460180_Q64/Diane-Guignard.jpg)
... We also allow for the possibility of plateaus in these functions. These degenerate elliptic equations hold significant relevance in various fields, particularly in porous media applications such as density-dependent groundwater flow modeling [17] and immiscible two-phase flow problems [39,40]. Additionally, they find crucial applications in biology [29,41] chemistry [55], and material sciences [7]. ...
![[object Object]](https://i1.rgstatic.net/ii/profile.image/788664208617472-1565043608175_Q64/Loic-Cappanera.jpg)
... While the numerical analysis of three-phase flow is sparse, we note that the case of immiscible two-phase flows in porous media has been investigated in several papers. For instance for incompressible flows, finite difference methods have been analyzed in [11], finite volume methods in [23,14,19], DG methods in [12], and finite element methods [9,15,16]. ...
... The convergence of numerical schemes proposed to approximate flows of incompressible fluids towards the (weak) solution of the original PDE problems is established in [22,27,61]. PDE analysis for a class of solids with bounded linearized strain is developed in [13,14,2], see also the survey paper [7], while a study regarding the properties of finite element approximations is presented in [4] (see also [28,37] for relevant results of computer simulations). Large data and long time PDE analysis of unsteady motions of generalized Kelvin-Voigt solids is developed in [12]; see also the study [9]. ...
... Examples include filter design, prosthetics, simulation of oil extraction from reservoirs, carbon sequestration, and sound insulation structures. From the viewpoint of constructing and analyzing numerical methods, recent works for the interfacial Biot/elasticity problem can be found in [4,5,[25][26][27]. These contributions include mortar-type discretizations, formulations using rotations, and extensions to lubrication models. ...
... Some of them have been extended to account for contact-mechanics as in [5] for the MSPA based on facewise constant approximations of the surface tractions and displacement jump along the fracture network, in [12] for HHO combined with a Nitsche's contact formulation, and in [30] for VEM based on node to node contact conditions. Among these polytopal methods, VEM, as a natural extension of the Finite Element Method (FEM) to polyhedral meshes, has received a lot of attention in the mechanics community since its introduction in [3] and has been applied to various problems including in the context of geomechanics [1], poromechanics [13,10,20] and fracture mechanics [31]. ...
