Gallouet Thierry

In the case of any bounded open set Ω ⊂ R d with boundary ∂Ω, we first construct a directional trace in any direction θ of the unit sphere, for any u ∈ L 2 (Ω) whose the directional derivative ∂ θ u in the direction θ belongs to L 2 (Ω). This directional trace is shown to belong to L 2 (∂Ω, µ θ), where µ θ is a measure supported by the closure of a...

We consider a finite volume scheme with two-point flux approximation (TPFA) to approximate a Laplace problem when the solution exhibits no more regularity than belonging to H 0 1 ( Ω ) H^1_0(\Omega ) . We define an error between the approximate solution and the exact one, involving their difference and the difference of normal gradients to the face...

We consider a finite volume scheme with two-point flux approximation (TPFA) to approximate a Laplace problem when the solution exhibits no more regularity than belonging to $H^1_0(\Omega)$. We establish in this case some error bounds for both the solution and the approximation of the gradient component orthogonal to the mesh faces. This estimate is...

We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the gradient discretisation method in space; the latter is in fact a class of methods that includes conforming, nonconforming and mixed finite element...

We consider a general linear parabolic problem with extended time boundary conditions (including initial value problems and periodic ones), and approximate it by the implicit Euler scheme in time and the Gradient Discretisation method in space; the latter is in fact a class of methods that includes conforming and nonconforming finite elements, disc...

The present paper addresses the convergence of a first-order in time incremental projection scheme for the time-dependent incompressible Navier–Stokes equations to a weak solution. We prove the convergence of the approximate solutions obtained by a semi-discrete scheme and a fully discrete scheme using a staggered finite volume scheme on non unifor...

The present paper addresses the convergence of the implicit Marker-and-Cell scheme for time-dependent Navier–Stokes equations with variable density and density-dependent viscosity and forcing term. A priori estimates on the unknowns are obtained, and thanks to a topological degree argument, they lead to the existence of an approximate solution at e...

We present a (partial) historical summary of the mathematical analysis of finite difference and finite volume methods, paying special attention to the Lax–Richtmyer and Lax–Wendroff theorems. We then state a Lax–Wendroff consistency result for convection operators on staggered grids (often used in fluid flow simulations), which illustrates a recent...

The present paper addresses the convergence of a first order in time incremental projection scheme for the time-dependent incompressible Navier-Stokes equations to a weak solution, without any assumption of existence or regularity assumptions on the exact solution. We prove the convergence of the approximate solutions obtained by the semi-discrete...

We consider the Stefan problem, firstly with regular data and secondly with irregular data. In both cases is given a proof for the convergence of an approximation obtained by regularising the problem. These proofs are based on weak formulations and on compactness results in some Sobolev spaces with negative exponents.

We prove in this paper the Lax–Wendroff consistency of a general finite volume convection operator acting on discrete functions which are possibly not piecewise-constant over the cells of the mesh and over the time steps. It yields an extension of the Lax–Wendroff theorem for general colocated or non-colocated schemes. This result is obtained for g...

The present paper is focused on the proof of the convergence of the discrete implicit Marker-and-Cell (MAC) scheme for time-dependent Navier--Stokes equations with variable density and variable viscosity. The problem is completed with homogeneous Dirichlet boundary conditions and is discretized according to a non-uniform Cartesian grid. A priori-es...

In this work we present a generic framework for non-conforming finite elements on polytopal meshes, characterised by elements that can be generic polygons/polyhedra. We first present the functional framework on the example of a linear elliptic problem representing a single-phase flow in porous medium. This framework gathers a wide variety of possib...

We present a (partial) historical summary of the mathematical analysis of finite differences and finite volumes methods, paying a special attention to the Lax-Richtmyer and Lax-Wendroff theorems. We then state a Lax-Wendroff consistency result for convection operators on staggered grids (often used in fluid flow simulations), which illustrates a re...

This paper is devoted to the numerical analysis of a numerical scheme dedicated to the simulation of front advection (see https://hal.archives-ouvertes.fr/hal-02940407v1 for a preprint version presenting this scheme and some numerical results). The latter has been recently proposed and it is based on the ideas used for the Glimm's scheme. It relies...

Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the internal energy equation, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either structured...

We prove in this paper the weak consistency of a general finite volume convection operator acting on discrete functions which are possibly not piecewise-constant over the cells of the mesh and over the time steps. It yields an extension of the Lax-Wendroff if-theorem for general colocated or non-colocated schemes. This result is obtained for genera...

A novel notion for constructing a well-balanced scheme – a gradient-robust scheme – is introduced and a showcase application for the steady compressible, isothermal Stokes equations in a nearly-hydrostatic situation is presented. Gradient-robustness means that gradient fields in the momentum balance are well-balanced by the discrete pressure gradie...

In this work we present a generic framework for non-conforming finite elements on polytopal meshes, characterised by elements that can be generic polygons/polyhedra. We first present the functional framework on the example of a linear elliptic problem representing a single-phase flow in porous medium. This framework gathers a wide variety of possib...

We propose in this paper a formally second order scheme for the numerical simulation of the shallow water equations in two space dimensions, based on the so-called Marker-And-Cell (MAC) staggered discretization on non uniform grids. For the space discretization, we use a MUSCL-like scheme for the convection operators while the pressure gadient is c...

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (con...

Numerical schemes for the solution of the Euler equations have recently been developed, which involve the discretisation of the internal energy equation, with corrective terms to ensure the correct capture of shocks, and, more generally, the consistency in the Lax-Wendroff sense. These schemes may be staggered or colocated, using either struc-tured...

The aim of this paper is to develop some tools in order to obtain the weak consistency of (in other words, an analogue of the Lax–Wendroff theorem for) finite volume schemes for balance laws in the multi-dimensional case and under minimal regularity assumptions for the mesh. As in the seminal Lax–Wendroff paper, our approach relies on a discrete in...

The aim of this paper is to develop some tools in order to obtain the weak consistency of (in other words, analogues of the Lax-Wendroff theorem for) finite volume schemes for balance laws in the multi-dimensional case and under minimal regularity assumptions for the mesh. As in the seminal Lax-Wendroff paper, our approach relies on a discrete inte...

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (con...

We adapt the Gradient Discretisation Method (GDM), originally designed for elliptic and parabolic partial differential equations, to the case of a linear scalar hyperbolic equations. This enables the simultaneous design and convergence analysis of various numerical schemes, corresponding to the methods known to be GDMs, such as finite elements (con...

We prove existence of a solution to the implicit MAC scheme for the compressible Navier–Stokes equations. We derive error estimates for this scheme on two and three dimensional Cartesian grids. Error estimates are obtained by using the discrete version of the relative energy method introduced on the continuous level in Feireisl et al. (J Math Fluid...

We show that three classical examples of schemes for the approximation of linear elliptic problems can be cast in a common framework, called the gradient discretisation method (GDM). An error estimate is then obtained by the extension to this framework of the second Strang lemma, which is completed by a second inequality showing that the conditions...

The GDM is presented, along with its convergence and error estimate properties, in the case of linear and quasi-linear elliptic problems with homogeneous and non-homogeneous Dirichlet boundary conditions.

A generic non-linear parabolic model which includes both Richards’ model describing the flow of water in a heterogeneous anisotropic underground medium, and Stefan’s model which arises in the study of a simplified heat diffusion in a melting medium.

Non-conforming methods are presented in the context of the GDM. An abstract framework is developed that covers a wide range of non-conforming methods, and the special case of non-conforming \(\mathbb {P}_k\) finite elements is then considered. In the case \(k=1\), the presentation is given for all classical boundary conditions, and mass-lumped non-...

Some non-degenerate parabolic problems are studied. An error estimate is obtained for a linear parabolic problem, followed by a convergence result for a quasi-linear problem. A class of non-linear parabolic problems under non-conservative form is then presented.

The GDM and its analysis are adapted here to cope with Neumann, Fourier and mixed boundary conditions. Properties of trace operators are detailed.

GDMs are built using the multi-point flux approximation-O scheme on rectangular and simplicial meshes.

The definition of gradient discretisations (GDs) for time-dependent problems is first given; it is followed by compactness results for the analysis of such problems.

Two GDMs are obtained from the Discontinuous Galerkin setting. The first one recovers the high order SIPG schemes in the case of linear problems, the second one, based on average jumps, leads to simpler computations.

GDMs are derived from mixed finite element schemes, using the initial and the hybrid formulations. High order estimates are proved in the case where the GDMs are issued from Raviart-Thomas mixed finite elements.

GDMs are obtained from the nodal mimetic finite differences methods, and also cover some DDFV schemes.

GDMs are constructed from the hybrid mimetic mixed schemes, recovering in particular the mimetic finite difference schemes and the SUSHI scheme.

Conforming Galerkin methods are shown to fit into the GDM. Emphasis is put on conforming \(\mathbb {P}_{k}\) finite elements, with precise estimates on their consistency and analysis for a variety of boundary conditions. Mass-lumped \(\mathbb {P}_{1}\) finite elements are also shown to be GDMs.

Analysis tools for GDM are presented. Polytopal toolboxes enable easy proofs of the coercivity, limit-conformity and compactness of gradient discretisations. The notion of local linearly exact gradient discretisations provides ways to analyse the consistency of GDs, as well as precise estimates on the consistency error.

After a brief review of a variety of discretisation methods for linear and non-linear elliptic problems, the basic ideas and motivations of GDM are presented.

We design an abstract setting for the approximation in Banach spaces of operators acting in duality. A typical example are the gradient and divergence operators in Lebesgue--Sobolev spaces on a bounded domain. We apply this abstract setting to the numerical approximation of Leray-Lions type problems, which include in particular linear diffusion. Th...

We present some discrete functional analysis tools for the proof of convergence of numerical schemes, mainly for equations including diffusion terms such as the Stefan problem or the Navier–Stokes equations in the incompressible and compressible cases. Some of the results covered here have been proved in previous works, coauthored with several cowo...

We prove in this paper the convergence of the marker and cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two- or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the ap...

In this paper, we derive entropy estimates for a class of schemes for the Euler equations which present the following features: they are based on the internal energy equation (eventually with a positive corrective term at the righ-hand-side so as to ensure consistency) and the possible upwinding is performed with respect to the material velocity on...

We establish an error estimate for fully discrete time-space gradient schemes on a simple linear parabolic equation. This error estimate holds for all the schemes within the framework of the gradient discretisation method: conforming and non conforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approxim...

We prove in this paper theconvergence of an semi-implicit MAC scheme for the time-dependent variable density Navier–Stokes equations.

The objective of this short paper is to present discrete functional analysis tools for proving the convergence of numerical schemes, mainly for elliptic and parabolic equations (Stefan problem and incompressible and compressible Navier–Stokes equations, for instance). The main part of these results are given in some papers coauthored with several c...

We prove in this paper the convergence of an semi-implicit MAC scheme for the time-dependent variable density Navier-Stokes equations.

We address in this paper a nonlinear parabolic system, which is built to retain the main mathematical difficulties of the P1 radiative diffusion physical model. We propose a finite volume fractional-step scheme for this problem enjoying the following properties. First, we show that each discrete solution satisfies a priori L -estimates, through a d...

We prove in this paper the convergence of the Marker and cell (MAC) scheme for the dis-cretization of the steady-state and unsteady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven ; they yie...

We prove in this paper the convergence of the Marker-and-Cell scheme for the discretization of the steady-state and time-dependent incompressible Navier–Stokes equations in primitive variables, on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven; they yield the...

We prove in this paper the convergence of the marker-and-cell (MAC) scheme for the discretization of the semi-stationary compressible Stokes equations on two or three dimensional Cartesian grids. Existence of a solution to the scheme is stated, followed by estimates on approximate solutions, which yields the convergence of the approximate solutions...

This monograph is dedicated to the presentation of the Gradient Discretisa- tion Method (GDM) and of some of its applications. It is intended for masters students, researchers and experts in the field of the numerical analysis of par- tial differential equations. The GDM is a framework which contains classical and recent discretization schemes for...

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three-dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the app...

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids.

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids.

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids.

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the app...

We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization of the steady state compressible and isentropic Navier-Stokes equations on two or three dimensional Cartesian grids. Existence of a solution to the scheme is proven, followed by estimates on approximate solutions, which yield the convergence of the app...

We consider a minimization problem of a functional in the space W01,p(Ω), where 1<p<+∞ and Ω is a bounded open set of RN. We prove the compactness, in the space W01,p(Ω), under convenient hypotheses, of a minimizing sequence. The main difficulty is to prove the convergence in measure of the gradient of the minimizing sequence. Furthermore, consider...

We study here the discretization by monotone finite volume schemes of multi-dimensional nonlinear scalar conservation laws forced by a multiplicative noise with a time and space dependent flux-function and a given initial data in $L^{2}(\R^d)$. After establishing the well-posedness theory for solutions of such kind of stochastic problems, we prove...

We show that the discrete operators and spaces of gradient discretizations can be designed so that the corresponding gradient scheme for a linear diffusion problem be identical to the Raviart-Thomas RTk mixed finite element method for both the primal mixed finite element formulation and the hybrid dual formulation. We then give the hybrid dual RT0...

This paper is devoted to the study of finite volume methods for the discretization of scalar conservation laws with a multiplicative stochastic force defined on a bounded domain D of R d with Dirichlet boundary conditions and a given initial data in L ∞ (D). We introduce a notion of stochastic entropy process solution which generalizes the concept...

We prove in this paper the convergence of the Marker and cell (MAC) scheme for the dis-cretization of the steady-state and unsteady-state incompressible Navier-Stokes equations in primitive variables on non-uniform Cartesian grids, without any regularity assumption on the solution. A priori estimates on solutions to the scheme are proven ; they yie...

We present here a general method based on the investigation of the relative energy of the system that provides an unconditional error estimate for the approximate solution of the barotropic Navier-Stokes equations obtained by time and space discretization. We use this methodology to derive an error estimate for a specific DG/finite element scheme f...

We address in this paper a non-linear parabolic system, which is built to retain the main mathematical difficulties of the P \(_1\) radiative diffusion physical model. We propose a finite volume fractional-step scheme for this problem enjoying the following properties. First, we show that each discrete solution satisfies a priori \({ L}^\infty \)-e...

In this paper, we prove the existence of a solution for a quite general stationary compressible Stokes problem including, in particular, gravity effects. The Equation Of State gives the pressure as an increasing superlinear function of the density. This existence result is obtained by passing to the limit on the solution of a viscous approximation...

We study here the discretization by monotone finite volume schemes of multi-dimensional nonlinear scalar conservation laws forced by a multiplicative noise with a time and space dependent flux-function and a given initial data in \(L^{2}(\mathbb {R}^d)\). After establishing the well-posedness theory for solutions of such kind of stochastic problems...

In this paper, we propose a discretization for the nonsteady compressible Stokes Problem. This scheme is based on Crouzeix-Raviart approximation spaces. The discretization of the momentum balance is obtained by the usual finite element technique. The discrete mass balance is obtained by a finite volume scheme, with an upwinding of the density. The...

We study here explicit flux-splitting finite volume discretizations of multi-dimensional nonlinear scalar conservation laws perturbed by a multiplicative noise with a given initial data in $L^{2}(\R^d)$. Under a stability condition on the time step, we prove the convergence of the finite volume approximation towards the unique stochastic entropy so...

A variational formulation of the standard MAC scheme for the approximation of the Navier-Stokes problem yields an extension of the scheme to general 2D and 3D domains and more general meshes. An original discretization of the trilinear form of the nonlinear convection term is proposed; it is designed so as to vanish for discrete divergence free fun...

Gradient schemes are nonconforming methods written in discrete variational formulation and based on independent approximations of functions and gradients, using the same degrees of freedom. Previous works showed that several well-known methods fall in the framework of gradient schemes. Four properties, namely coercivity, consistency, limit-conformi...

We give in this paper a short review of some recent achievements within the framework of multiphase flow modeling. We focus first on a class of compressible two-phase flow models, detailing closure laws and their main properties. Next we briefly summarize some attempts to model two-phase flows in a porous region, and also a class of compressible th...

In this paper, we propose a discretization for the compressible Stokes problem with an equation of state of the form p=φ(ρ) (where p stands for the pressure and ρ for the density, and φ is a superlinear nondecreasing function from ℝ to ℝ). This scheme is based on Crouzeix–Raviart approximation spaces. The discretization of the momentum balance is o...

This special issue of IJFV gathers some invited contributions that are devoted to the mathematical and numerical and physical modelling of two-phase flows. This topic is a motivating and thriving research area, with numerous industrial applications including, among others, the prediction of fluid flows in turbines, pressurized water reactors, and s...

L’objet de ce livre est de donner une vue d’ensemble de la théorie de la mesure, de l’intégration et des probabilités correspondant à un niveau de troisième année de licence ou de première année de master (en mathématiques).La lecture de ce livre requiert la connaissance des notions d’analyse réelle, d’algèbre linéaire et de calcul différentiel ens...

We give here a comparative study on the mathematical analysis of two (classes of) discretization schemes for the computation of approximate solutions to incompressible two-phase flow problems in homogeneous porous media. The first scheme is the well-known finite volume scheme with a two-point flux approximation, classically used in industry. The se...

We show in this paper that the gradient schemes (which encompass a large family of discrete schemes) may be used for the approximation of the Stefan problem $\partial_t \bar u - \Delta \zeta (\bar u) = f$. The convergence of the gradient schemes to the continuous solution of the problem is proved thanks to the following steps. First, estimates show...

In this paper we study the existence of W01,1(Ω) distributional solutions of Dirichlet problems whose simplest example is{−div(|∇u|p−2∇u)=f(x),in Ω;u=0,on ∂Ω.

Finite volume schemes for the approximation of a biharmonic problem with Dirichlet boundary conditions are constructed and analyzed, first on grids which satisfy an orthogonality condition, and then on general, possibly non conforming meshes. In both cases, the piece-wise constant approximate solution is shown to converge in L2 () to the exact solu...

In this paper, we are interested in modeling single-phase flow in a porous medium with known faults seen as interfaces. We mainly focus on how to handle non-matching grids problems arising from rock displacement along the fault. We describe a model that can be extended to multi-phase flow where faults are treated as interfaces. The model is validat...

We prove in this paper the continuity of the natural projection operator from W 01,q (Ω) d, q ∈ [1, + ∞), d = 2 or d = 3, to the MAC discrete space of piecewise constant functions over the dual cells, endowed with the finite volume W 01,q-discrete norm. Since this projection operator is also a Fortin operator (that is an operator which "preserves"...

Gradient schemes are nonconforming methods written in discrete variational formulation and based on independent approximations of functions and gradients, using the same degrees of freedom. Previous works showed that several well-known methods fall in the framework of gradient schemes. Four properties, namely coercivity, consistency, limit-conformi...

In this paper, we prove the existence of weak solutions for mathematical models of miscible and immiscible flow through porous medium. An important difficulty comes from the modelization of the wells, which does not allow us to use classical variational formulations of the equations.

In this paper, we prove an adaptation of the classical compactness Aubin-Simon lemma to sequences of functions obtained through a sequence of discretizations of a parabolic problem. The main difficulty tackled here is to generalize the classical proof to handle the dependency of the norms controlling each function u (n) of the sequence with respect...

We show that any entropy solution u of a convection diffusion equation ∂tu+divF(u)−∆φ(u) = b in Ω×(0, T) belongs to C([0, T),L 1 loc(Ω)). The proof does not use the uniqueness of the solution. 1 The problem, and main result Convection diffusion equations appear in a large class of problems, and have been widely studied. We consider in the sequel on...

In this paper, we propose a discretization for the compressible Stokes problem with an equation of state of the form p = φ(ρ) (where p stands for the pressure, ρ for the density and φ is a nondecreasing function belonging to (_+, ){C}^{1}({\mathbb{R}}_{+}, \mathbb{R}) ). This scheme is based on Crouzeix-Raviart approximation spaces. The discretiz...

The 1D Burgers equation is used as a toy model to mimick the resulting behaviour of numerical schemes when replacing a conservation law by a form which is equivalent for smooth solutions, such as the total energy by the internal energy balance in the Euler equations. If the initial Burgers equation is replaced by a balance equation for one of its e...