Papers by Thierry Gallouët
HAL (Le Centre pour la Communication Scientifique Directe), Jun 8, 2008
Ima Journal of Numerical Analysis, Jun 16, 2009
A symmetric discretisation scheme for heterogeneous anisotropic diffusion problems on general mes... more A symmetric 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. Convergence of the approximate solution to the continuous solution is proved for general (possibly discontinuous) tensors, general (possibly nonconforming) meshes, and with no regularity assumption on the solution. An error estimate is then deduced under suitable regularity assumptions on the solution.
arXiv (Cornell University), Jul 21, 2021
The present paper is focused on the proof of the convergence of the discrete implicit Marker-and-... more 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. Finite volume methods; MAC scheme; incompressible Navier-Stokes equations; variable density and viscosity; transport equations.
Numerical Simulation in Physics and Engineering: Trends and Applications, 2021
Polyhedral Methods in Geosciences, 2021
In this work we present a generic framework for non-conforming finite elements on polytopal meshe... more 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 possible nonconforming methods, and an error estimate is provided for this simple model. We then turn to the application of the functional framework to the case of a steady degenerate elliptic equation, for which a mass-lumping technique is required; here, this technique simply consists in using a different-piecewise constant-function reconstruction from the chosen degrees of freedom. A convergence result is stated for this degenerate model. Then, we introduce a novel specific non-conforming method, dubbed Locally Enriched Polytopal Non-Conforming (LEPNC). These basis functions comprise functions dedicated to each face of the mesh (and associated with average values on these faces), together with functions spanning the local P 1 space in each polytopal element. The analysis of the interpolation properties of these basis functions is provided, and mass-lumping techniques are presented. Numerical tests are presented to assess the efficiency and the accuracy of this method on various examples. Finally, we show that generic polytopal non-conforming methods, including the LEPNC, can be plugged into the gradient discretization method framework, which makes them amenable to all the error estimates and convergence results that were established in this framework for a variety of models.
The present paper is focused on the proof of the convergence of the discrete implicit Marker-andC... more The present paper is focused on the proof of the convergence of the discrete implicit Marker-andCell (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. Finite volume methods; MAC scheme; incompressible Navier–Stokes equations; variable density and viscosity; transport equations.
arXiv: Numerical Analysis, 2017
In this paper, we derive entropy estimates for a class of schemes for the Euler equations which p... more 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 only. The implicit-in-time first-order upwind scheme satisfies a local entropy inequality. A generalization of the convection term is then introduced, which allows to limit the scheme diffusion while ensuring a weaker property: the entropy inequality is satisfied up to a remainder term which is shown to tend to zero with the space and time steps, if the discrete solution is controlled in L ∞ and BV norms. The explicit upwind variant also satisfies such a weaker property, at the price of an estimate for the velocity which could be derived from the introduction of a new stabilization term in the momentum balance. Still for the explicit scheme,...
Computer Methods in Applied Mechanics and Engineering, 2020
A novel notion for constructing a well-balanced scheme-a gradient-robust scheme-is introduced and... more A novel notion for constructing a well-balanced scheme-a gradient-robust scheme-is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient-if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM-FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straightforward extension to barotropic situations with nonlinear equations of state is feasible.
Mathematics of Computation, 2017
We prove in this paper the convergence of the Marker and Cell (MAC) scheme for the discretization... more 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 approximate solutions, up to a subsequence, and in an appropriate sense. We then prove that the limit of the approximate solutions satisfies the mass and momentum balance equations, as well as the equation of state, which is the main difficulty of this study.
IMA Journal of Numerical Analysis, 2015
On one hand, the existence of a solution to degenerate parabolic equations, without a nonlinear c... more On one hand, the existence of a solution to degenerate parabolic equations, without a nonlinear convection term, can be proven using the results of Alt and Luckhaus, Minty and Kolmogorov. On the other hand, the proof of uniqueness of an entropy weak solution to a nonlinear scalar hyperbolic equation, first provided by Krushkov, has been extended in two directions: Carrillo has handled the case of degenerate parabolic equations including a nonlinear convection term, whereas Di Perna has proven the uniqueness of weaker solutions, namely Young measure entropy solutions. All of these results are reviewed in the course of a convergence result for two regularizations of a degenerate parabolic problem including a nonlinear convective term. The first regularization is classicaly obtained by adding a minimal diffusion, the second one is given by a finite volume scheme on unstructured meshes. The convergence result is therefore only based on L ∞ (Ω × (0, T)) and L 2 (0, T ; H 1 (Ω)) estimates, associated with the uniqueness result for a weaker sense for a solution.
Computational and Applied Mathematics, 2014
ESAIM: Proceedings, 2013
We give in this paper a short review of some recent achievements within the framework of multipha... more 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 three-phase flow models. Some of the main difficulties arising in the numerical simulation of solutions of these complex and highly non-linear systems of PDEs are then discussed, and we eventually show some numerical results when tackling twophase flows with mass transfer. Résumé. Quelques résultats concernant la modélisation desécoulements multiphasiques Nous présentons dans cet article quelques résultats récents concernant la modélisation et la simulation numérique desécoulements multiphasiques. Nous nous concentrons tout d'abord sur une classe de modèles diphasiques compressibles, en détaillant les lois de fermeture et les principales propriétés du sytème. Nous résumons ensuite brièvement les propositions de modélisation d'écoulements diphasiques en milieu poreux et d'écoulements triphasiques. Quelques difficultés apparaissant dans la simulation numérique de ces modèles sont présentées, et des résultats récents comportant un transfert de masse entre phases sont finalement décrits.
Annales de la faculté des sciences de Toulouse Mathématiques, 2014
SIAM Journal on Numerical Analysis, 2010
Mathematics of Computation, 2012
In this paper, we prove the convergence of a finite-volume scheme for the time-dependent convecti... more In this paper, we prove the convergence of a finite-volume scheme for the time-dependent convection-diffusion equation with an L 1 right-hand side. To this purpose, we first prove estimates for the discrete solution and for its discrete time and space derivatives. Then we show the convergence of a sequence of discrete solutions obtained with more and more refined discretizations, possibly up to the extraction of a subsequence, to a function which meets the regularity requirements of the weak formulation of the problem; to this purpose, we prove a compactness result, which may be seen as a discrete analogue to the Aubin-Simon lemma. Finally, such a limit is shown to be indeed a weak solution.
Mathematical Models and Methods in Applied Sciences, 2013
Gradient schemes are nonconforming methods written in discrete variational formulation and based ... more 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-conformity and compactness, are shown in this paper to be sufficient to prove the convergence of gradient schemes for linear and nonlinear elliptic and parabolic problems, including the case of nonlocal operators arising for example in image processing. We also show that the schemes of the Hybrid Mimetic Mixed family, which include in particular the Mimetic Finite Difference schemes, may be seen as gradient schemes meeting these four properties, and therefore converges for the class of above-mentioned problems.
Journal of Numerical Mathematics, 2009
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Papers by Thierry Gallouët