Robert Eymard

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• Professor
• Professor at Gustave Eiffel University

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 first give a general error estimate for the nonconforming approximation of a problem for which a Banach–Nečas–Babuška (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types...

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 prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier–Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretized in conforming spaces, whose compatibility is ensured by the existence of an interpolator for regular functions whi...

We define, in the case of quasilinear convection-diffusion equations, an approximation of the numerical fluxes obtained by extending the Scharfetter and Gummel fluxes (defined in the case of linear convection—diffusion). We show that this approximation is compatible with the asymptotic thermal equilibrium on an application example.

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...

We first give a general error estimate for the nonconforming approximation of a problem for which a Banach-Nečas-Babuška (BNB) inequality holds. This framework covers parabolic problems with general conditions in time (initial value problems as well as periodic problems) under minimal regularity assumptions. We consider approximations by two types...

We prove the convergence of an incremental projection numerical scheme for the time-dependent incompressible Navier--Stokes equations, without any regularity assumption on the weak solution. The velocity and the pressure are discretised in conforming spaces, whose the compatibility is ensured by the existence of an interpolator for regular function...

Given a densely defined skew-symmetric operator $$A_0$$ A 0 on a real or complex Hilbert space V , we parameterize all m -dissipative extensions in terms of contractions $$\Phi :{H_{{-}}}\rightarrow {H_{{+}}}$$ Φ : H - → H + , where $${H_{{-}}}$$ H - and $${H_{{+}}}$$ H + are Hilbert spaces associated with a boundary quadruple . Such an extension g...

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...

We propose a new numerical 2-point flux for a quasilinear convection-diffusion equation. This numerical flux is shown to be an approximation of the numerical flux derived from the solution of a two-point Dirichlet boundary value problem for the projection of the continuous flux onto the line connecting neighboring collocation points. The later appr...

We approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weigh...

Given a densely defined skew-symmetric operators A 0 on a real or complex Hilbert space V , we parametrize all m-dissipative extensions in terms of contractions $\Phi$ : H-$\rightarrow$ H + , where Hand H + are Hilbert spaces associated with a boundary quadruple. Such an extension generates a unitary C 0-group if and only if $\Phi$ is a unitary ope...

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 approximate the solution to some linear and degenerate quasi-linear problem involving a linear elliptic operator (like the semi-discrete in time implicit Euler approximation of Richards and Stefan equations) with measure right-hand side and heterogeneous anisotropic diffusion matrix. This approximation is obtained through the addition of a weigh...

The Representation Theorem of Lions (RTL) is a version of the Lax--Milgram Theorem where completeness of one of the spaces is not complete. In this paper, RTL is deduced from an operator-theoretical version on normed space. The main point of the paper is a theory of derivations, based on RTL, for which well-posedness is proved. One application conc...

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.

This work is devoted to the study of the approximation, using two nonlinear numerical methods, of a linear elliptic problem with measure data and heterogeneous anisotropic diffusion matrix. Both methods show convergence properties to a continuous solution of the problem in a weak sense, through the change of variable u = ψ(v), where ψ is a well cho...

In this paper, we apply the characteristic finite volume method (CFVM) for solving a convection-diffusion problem on two-dimensional triangular grids. The finite volume method is used to discretize the equation while the finite element method is applied to estimate the gradient quantities at cell faces. The numerical analysis of the convergence has...

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...

We show that, using the Crouzeix-Raviart scheme, a cheap algebraic transformation, applied to the coupled velocity-pressure linear systems issued from the transient or steady Stokes or Navier-Stokes problems, leads to a linear system only involving as many auxiliary variables as the velocity components. This linear system, which is symmetric positi...

In this paper we study the conforming Galerkin approximation of the problem: find $u\in{{\mathcal{U}}}$ such that $a(u,v) = \langle L, v \rangle $ for all $v\in{{\mathcal{V}}}$, where ${{\mathcal{U}}}$ and ${{\mathcal{V}}}$ are Hilbert or Banach spaces, $a$ is a continuous bilinear or sesquilinear form and $L\in{{\mathcal{V}}}^{\prime}$ a given dat...

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...

Revision of the previous version, including a development on saddle-point problems

Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite Volume methods. We show here that this framework can be applied to a family of degenerate non-linear paraboli...

In this paper, we present a class of finite volume schemes for incompressible flow problems. The unknowns are collocated at the center of the control volumes, and the stability of the schemes is obtained by adding to the mass balance stabilization terms involving the pressure jumps across the edges of the mesh.

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...

The approximation of problems with linear convection and degenerate nonlinear diffusion, which arise in the framework of the transport of energy in porous media with thermodynamic transitions, is done using a θ-scheme based on the centred gradient discretisation method. The convergence of the numerical scheme is proved, although the test functions...

In this paper we study the conforming Galerkin approximation of the problem: find u $\in$ U such that a(u, v) = L, v for all v $\in$ V, where U and V are Hilbert or Banach spaces, a is a continuous bilinear or sesquilinear form and L $\in$ V a given data. The approximate solution is sought in a finite dimensional subspace of U, and test functions a...

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 present and analyse a numerical framework for the approximation of nonlinear degenerate elliptic equations of the Stefan or porous medium types. This framework is based on piecewise constant approximations for the functions, which we show are essentially necessary to obtain convergence and error estimates. Convergence is established without regu...

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 include in the gradient discretisation method (GDM) framework two numerical schemes based on discontinuous Galerkin approximations: the symmetric interior penalty Galerkin (SIPG) method, and the scheme obtained by averaging the jumps in the SIPG method. We prove that these schemes meet the main mathematical gradient discretisation properties on...

This paper includes an erratum to (Bouchut et al. (2014) Convection and total variation flow. IMA J. Numer. Anal., 34, 1037-1071.) which deals with a nonlinear hyperbolic scalar conservation law, regularized by the total variation flow operator (or 1-Laplacian), and in which a mistake occurred in the convergence proof of the numerical scheme to the...

This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a...

This article performs a unified convergence analysis of a variety of numerical methods for a model of the miscible displacement of one incompressible fluid by another through a porous medium. The unified analysis is enabled through the framework of the gradient discretisation method for diffusion operators on generic grids. We use it to establish a...

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 show that a version of the Discontinuous Galerkin Method (DGM) can be included in the Gradient Discretisation Method (GDM) framework. We prove that it meets the main mathematical gradient discretisation properties on any kind of polytopal mesh, and that it is identical to the Symmetric Interior Penalty Galerkin (SIPG) method in the case of first...

An asymmetric version of the gradient discretisation method is developed for linear anisotropic elliptic equations. Error estimates and convergence are proved for this method, which is showed to cover all finite volume methods.

We recall theextension of the Marker and Cell (MAC) scheme for locally refined grids which was introduced in Chénier et al. (Calcolo 52(1), 69–107 (2015), [3]) and present the results obtained on the lid driven cavity test.

We brieflypresent the Hybrid Mimetic Mixed scheme for the steady incompressible Navier–Stokes equations. Two centred approximations of the nonlinear convection term are proposed and compared, between themselves as well as with reference results from the literature, on the lid driven cavity test case applied to various grid types.

This paper presents the common mathematical features which are leading to convergence prop- erties for a family of numerical schemes applied to the discretisation of the steady and transient incompressible Navier-Stokes equations with homogeneous Dirichlet’s boundary conditions. This family includes the Taylor-Hood scheme, the MAC scheme, the Crouz...

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 consider a time-dependent and a steady linear convection-diffusion-reaction equation whose coefficients are nonconstant. Boundary conditions are mixed (Dirichlet and Robin-Neumann) and nonhomogeneous. Both the unsteady and the steady problem are approximately solved by a combined finite element – finite volume method: the diffusion term is discr...

We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic p-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original...

We study the behaviour of solutions to a class of nonlinear degenerate parabolic problems when the data are perturbed. The class includes the Richards equation, Stefan problem and the parabolic p-Laplace equation. We show that, up to a subsequence, weak solutions of the perturbed problem converge uniformly-in-time to weak solutions of the original...

The staggered numerical scheme is shown to be a robust and simple method for the approximation of the Exner - shallow water equations for bedload sediment modelling. Numerical tests show good convergence properties to an analytical solution and match pretty well data experiments in the case of dam break with erodible bottom. The cases of subcritica...

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...

Gradient schemes is a framework that enables the unified convergence analysis of many numerical methods for elliptic and parabolic partial differential equations: conforming and non-conforming Finite Element, Mixed Finite Element and Finite Volume methods. We show here that this framework can be applied to a family of degenerate non-linear paraboli...

We show that solutions to a class of nonlinear degenerate parabolic initial-boundary value problems exhibit uniform temporal stability when the coefficients and data are perturbed. The class of equations encompasses the Richards model of groundwater flow, the Stefan problem and the parabolic $p$-Laplace equation (or, more generally, parabolic Leray...

Paris 6, F-75005 Paris, and INRIA, ANGE project-team, Rocquencourt-B.P. 105 F78153 Le Chesnay cedex, and CEREMA, ANGE project-team, Abstract. We apply the Gradient Schemes framework to the approximation of the incompressible steady Navier-Stokes problem. We show that some classical schemes (Crouzeix-Raviart, conforming Taylor-Hood and MAC) enter in...

The gradient scheme framework is based on a small number of properties and encompasses a large number of numerical methods for diffusion models. We recall these properties and develop some new generic tools associated with the gradient scheme framework. These tools enable us to prove that classical schemes are indeed gradient schemes, and allow us...

We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergenc...

The gradient scheme framework encompasses several conforming and non-conforming numerical schemes for diffusion equations. We develop here this framework for the approximation of the steady state and transient incompressible Stokes equations with homogeneous Dirichlet boundary conditions. Using this framework, we establish generic convergence resul...

We consider a time-dependent and a steady linear convection-diffusion equation. These equations are approximately solved by a combined finite element -- finite volume method: the diffusion term is discretized by Crouzeix-Raviart piecewise linear finite elements on a triangular grid, and the convection term by upwind barycentric finite volumes. In t...

We study approximations by conforming methods of the solution to the variational inequality $\langle \partial_t u,v-u\rangle + \psi(v) - \psi(u) \ge \langle f,v-u\rangle$, which arises in the context of inviscid incompressible Bingham fluid flows and of the total variation flow problem. We propose a general framework involving total variation funct...

The gradient scheme family, which includes the conforming and mixed finite elements as well as the mimetic mixed hybrid family, is used for the approximation of Richards equation and the two-phase flow problem in heterogeneous porous media. We prove the convergence of the approximate saturation and of the approximate pressures and approximate press...

The flow of a Bingham fluid with inertial terms is simplified into a nonlinear hyperbolic scalar conservation law, regularised by the total variation flow operator (or 1-Laplacian). We give an entropy weak formulation, for which we prove the existence and the uniqueness of the solution. The existence result is established using the convergence of a...

We study the convergence of two generalized marker‐and‐cell covolume schemes for the incompressible Stokes and Navier–Stokes equations introduced by Cavendish, Hall, Nicolaides, and Porsching. The schemes are defined on unstructured triangular Delaunay meshes and exploit the Delaunay–Voronoi duality. The study is motivated by the fact that the rela...

Problem The role of mangroves (coastal forests) in the mitigation of tsunami impacts is a debated topic and numerical simulation can bring an important contribution to this debate. Several articles have been devoted to the numerical modelling of tsunami mitigation by mangroves in the past few years [1, 2, 3]. The arrival of a tsunami in a coastal a...

We focus here on the difficult problem of linear solving, when considering implicit scheme for two-phase flow simulation in porous media. Indeed, this scheme leads to ill-conditioned linear systems, due to the different behaviors of the pressure unknown (which follows a diffusion equation) and the saturation unknown (mainly advected by the total vo...

We prove that all Gradient Schemes—which include Finite Element, Mixed Finite Element, Finite Volume methods—converge uniformly in time when applied to a family of nonlinear parabolic equations which contains Richards and Stefan’smodels.We also provide numerical results to confirm our theoretical analysis.

We study the convergence of a finite volume scheme for a model of miscible two-phase flow in porous media. In this model, one phase can dissolve into the other one. The convergence of the scheme is proved thanks to an estimate on the two pressures, which allows to prove some estimates on the discrete time derivative of some nonlinear functions of t...

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...

We present a new formulation of piecewise deterministic Markov processes with boundary. We first introduce a generalization of the Kolmogorov equations to which the marginal distributions of the process are the solution, and we prove the uniqueness of this solution. We then propose a finite volume numerical scheme for its approximation, and we prov...

Comparison exercises have been carried out by different research teams to study the sensitivity of the natural convection occurring in a vertical asymmetrically heated channel to four sets of open boundary conditions. The dimensionless parameters have been chosen so that a return flow exists at the outlet. On the whole, results provided by the part...

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...

Key words: Porous medium equation, immiscible gas-liquid Darcy flow in porous media, diffusion-migration of ions in porous media. Abstract. The porous medium equation, which originally describes the conservation of a gas injected in a homogeneous porous medium, is shown in this paper to also lead the main features of some models of mass transfer wi...