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Thierry Gallouët

Publications and source records attributed to Thierry Gallouët.

17 recordsLinked to original sources
arXiv2026

The omnidirectional trace in H 1 ($Ω$)

We first prove that all the functions in L 2 whose directional derivative is in L 2 have a directional trace on the boundary of any open bounded domain, without assumptions on its regularity. This enables us to define the omnidirectional trace of the elements of the Sobolev space H 1 ($Ω$) for which there exists a function on the boundary that is almost everywhere equal, with respect to the directional measure, to the directional trace, regardless of the direction. The set of all these elements of H 1 ($Ω$), denoted by H 1 tr ($Ω$), is shown to be closed, and to always contain the closure in H 1 ($Ω$) of the set C 0 ($Ω$)$\cap$ H 1 ($Ω$) (it is always equal to this set in the 1D case, and can be strictly greater in higher dimensions). The omnidirectional trace always satisfies an integration-by-parts formula, which combines the values of the trace on opposite points of the boundary. Examples show that this notion enables the resolution of variational problems involving the values at the boundary of the domain.

math.AP↗
arXiv2026

Trace operator on H 1 ($Ω$) for general open bounded domains

In the case of any bounded open set $Ω$ $\subset$ R d with boundary $\partial$$Ω$, we first construct a directional trace in any direction $θ$ of the unit sphere, for any u $\in$ L 2 ($Ω$) whose the directional derivative $\partial$ $θ$ u in the direction $θ$ belongs to L 2 ($Ω$). This directional trace is shown to belong to L 2 ($\partial$$Ω$, $μ$ $θ$ ), where $μ$ $θ$ is a measure supported by the closure of all points of $\partial$$Ω$ which are the extremity of an open segment directed by $θ$, included in $Ω$. This trace enables an integration by parts formula. We then show that the set H 1 tr ($Ω$) containing the elements of H 1 ($Ω$) whose the directional trace does not depend on $θ$ is closed. It therefore contains the closure of H 1 ($Ω$) $\cap$ C 0 ($Ω$) in H 1 ($Ω$). Examples where H 1 tr ($Ω$) = H 1 ($Ω$) and H 1 tr ($Ω$) __ = H 1 ($Ω$) are provided.

math.FA↗
arXiv2024

A SIR epidemic model on a refining spatial grid II-Central limit theorem

A stochastic SIR epidemic model taking into account the heterogeneity of the spatial environment is constructed. The deterministic model is given by a partial differential equation and the stochastic one by a space-time jump Markov process. The consistency of the two models is given by a law of large numbers. In this paper, we study the deviation of the spatial stochastic model from the deterministic model by a functional central limit theorem. The limit is a distribution-valued Ornstein-Uhlenbeck Gaussian process, which is the mild solution of a stochastic partial differential equation.

math.PR↗
arXiv2024

Optimal error bounds for the two point flux approximation finite volume scheme

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 $H^1_0(Ω)$. 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 optimal, in the sense that the approximation error has the same order as that of the sum of the interpolation error and a conformity error. A numerical example illustrates the error estimate in the context of a solution with minimal regularity. This result is extended to evolution problems discretized via the implicit Euler scheme in an appendix.

math.NA↗
arXiv2022

Convergence of the fully discrete incremental projection scheme for incompressible flows

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 scheme and a fully discrete scheme using a staggered finite volume scheme on non uniform rectangular meshes. Some first a priori estimates on the approximate solutions yield the existence. Compactness arguments, relying on these estimates, together with some estimates on the translates of the discrete time derivatives, are then developed to obtain convergence (up to the extraction of a subsequence), when the time step tends to zero in the semi-discrete scheme and when the space and time steps tend to zero in the fully discrete scheme; the approximate solutions are thus shown to converge to a limit function which is then shown to be a weak solution to the continuous problem by passing to the limit in these schemes.

math.NA↗
arXiv2022

A new convergence proof for approximations of the Stefan problem

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.

math.NA↗
arXiv2021

Convergence of the implicit MAC-discretized Navier--Stokes equations with variable density and viscosity on non-uniform grids

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-estimates on the unknowns are obtained, and along with a topological degree argument they lead to the existence of a solution of the discrete scheme at each time step. We conclude with the proof of the convergence of the scheme toward the continuous problem as mesh size and time step tend toward zero with the limit of the sequence of discrete solutions being a solution to the weak formulation of the problem.

math.NA↗
arXiv2019

On The Weak Consistency of Finite Volumes Schemes for Conservation Laws on General Meshes

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 integration by parts of the weak formulation of the scheme. This makes a discrete gradient of the test function appear, and the central argument for the scheme consistency is to remark that this discrete gradient is convergent in L $\infty$ weak .

math.NA↗
arXiv2017

Analysis of a fractional-step scheme for the P1 radiative diffusion model

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 discrete maxi- mum principle; by a topological degree argument, this yields the existence of a solution, which is proven to be unique. Second, we establish uniform (with respect to the size of the meshes and the time step) L2 -bounds for the space and time translates; this proves, by the Kolmogorov theorem, the relative compactness of any sequence of solutions obtained through a sequence of discretizations the time and space steps of which tend to zero; the limits of converging subsequences are then shown to be a solution to the continuous problem. Estimates of time translates of the discrete solutions are obtained through the formalization of a generic argument, interesting for its own sake.

math.NA↗
arXiv2017

Convergence of the MAC scheme for the incompressible Navier-Stokes equations

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 yield the existence of discrete solutions and the compactness of sequences of solutions obtained with family of meshes the space step of which tends to zero. We then establish that the limit is a weak solution to the continuous problem.

math.NA↗
arXiv2010

Some possibly degenerate elliptic problems with measure data and non linearity on the boundary

The goal of this paper is to study some possibly degenerate elliptic equation in a bounded domain with a nonlinear boundary condition involving measure data. We investigate two types of problems: the first one deals with the laplacian in a bounded domain with measure supported on the domain and on the boundary. A second one deals with the same type of data but involves a degenerate weight in the equation. In both cases, the nonlinearity under consideration lies on the boundary. For the first problem, we prove an optimal regularity result, whereas for the second one the optimality is not guaranteed but we provide however regularity estimates.

math.AP↗
arXiv2008

A unified approach to Mimetic Finite Difference, Hybrid Finite Volume and Mixed Finite Volume methods

We investigate the connections between several recent methods for the discretization of anisotropic heterogeneous diffusion operators on general grids. We prove that the Mimetic Finite Difference scheme, the Hybrid Finite Volume scheme and the Mixed Finite Volume scheme are in fact identical up to some slight generalizations. As a consequence, some of the mathematical results obtained for each of the method (such as convergence properties or error estimates) may be extended to the unified common framework. We then focus on the relationships between this unified method and nonconforming Finite Element schemes or Mixed Finite Element schemes, obtaining as a by-product an explicit lifting operator close to the ones used in some theoretical studies of the Mimetic Finite Difference scheme. We also show that for isotropic operators, on particular meshes such as triangular meshes with acute angles, the unified method boils down to the well-known efficient two-point flux Finite Volume scheme.

math.NA↗
arXiv2008

Discretisation of heterogeneous and anisotropic diffusion problems on general non-conforming meshes. SUSHI: a scheme using stabilisation and hybrid interfaces

A discretisation scheme for heterogeneous anisotropic diffusion problems on general meshes is developed and studied. The unknowns of this scheme are the values at the centre of the control volumes and at some internal interfaces which may for instance be chosen at the diffusion tensor discontinuities. The scheme is therefore completely cell centred if no edge unknown is kept. It is shown to be accurate on several numerical examples. Mathematical convergence of the approximate solution to the continuous solution is obtained for general (possibly discontinuous) tensors, general (possibly non-conforming) meshes, and with no regularity assumption on the solution. An error estimate is then drawn under sufficient regularity assumptions on the solution.

math.NA↗
arXiv2008

A convergent Finite Element-Finite Volume scheme for the compressible Stokes problem Part I -- the isothermal case

In this paper, we propose a discretization for the (nonlinearized) compressible Stokes problem with a linear equation of state $ρ=p$, based on Crouzeix-Raviart elements. The approximation of the momentum balance is obtained by usual finite element techniques. Since the pressure is piecewise constant, the discrete mass balance takes the form of a finite volume scheme, in which we introduce an upwinding of the density, together with two additional stabilization terms. We prove {\em a priori} estimates for the discrete solution, which yields its existence by a topological degree argument, and then the convergence of the scheme to a solution of the continuous problem.

math.NA↗
arXiv2007

An unconditionnally stable pressure correction scheme for compressible barotropic Navier-Stokes equations

We present in this paper a pressure correction scheme for barotropic compressible Navier-Stokes equations, which enjoys an unconditional stability property, in the sense that the energy and maximum-principle-based a priori estimates of the continuous problem also hold for the discrete solution. The stability proof is based on two independent results for general finite volume discretizations, both interesting for their own sake: the $L^2$-stability of the discrete advection operator provided it is consistent, in some sense, with the mass balance and the estimate of the pressure work by means of the time derivative of the elastic potential. The proposed scheme is built in order to match these theoretical results, and combines a fractional-step time discretization of pressure-correction type to a space discretization associating low order non-conforming mixed finite elements and finite volumes. Numerical tests with an exact smooth solution show the convergence of the scheme.

math.NA↗
arXiv2005

A cell-centred finite volume approximation for second order partial derivative operators with full matrix on unstructured meshes in any space dimension

Finite volume methods for problems involving second order operators with full diffusion matrix can be used thanks to the definition of a discrete gradient for piecewise constant functions on unstructured meshes satisfying an orthogonality condition. This discrete gradient is shown to satisfy a strong convergence property on the interpolation of regular functions, and a weak one on functions bounded for a discrete $H^1$ norm. To highlight the importance of both properties, the convergence of the finite volume scheme on a homogeneous Dirichlet problem with full diffusion matrix is proven, and an error estimate is provided. Numerical tests show the actual accuracy of the method.

math.NA↗