# Pascal OmnesAtomic Energy and Alternative Energies Commission | CEA · Centre d'Etudes de Saclay

Pascal Omnes

Habilitation

A posteriori error estimation, multiscale finite elements, domain decomposition for fluid flows

## About

47

Publications

11,018

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1,058

Citations

Citations since 2017

Introduction

Additional affiliations

September 2007 - present

February 1999 - present

December 1996 - March 1998

## Publications

Publications (47)

We propose and analyze a new parallel paradigm that uses both the time and the space directions. The original approach couples the Parareal algorithm with incomplete optimized Schwarz waveform relaxation (OSWR) iterations. The analysis of this coupled method is presented for a one-dimensional advection-reaction-diffusion equation. We also prove a g...

In this article, we consider the time dependent convection--diffusion--reaction equation coupled with the Darcy equation. We propose a numerical scheme based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. We establish optimal a posteriori error estimates with two types of comp...

In this paper, we study in two and three space dimensions, the a posteriori error estimates for the Large Eddy Simulation applied to the Navier-Stokes system. We begin by introducing the Navier-Stokes and the corresponding Large Eddy Simulation (LES) equations. Then we introduce the corresponding discrete problem based on the finite element method....

Classical finite volume schemes for the Euler system are not accurate at low Mach number and some fixes have to be used and were developed in a vast literature over the last two decades. The question we are interested in in this article is: What about if the porosity is no longer uniform? We first show that this problem may be understood on the lin...

In the aim to find the simplest and most efficient shape of a noise absorbing wall to dissipate the acoustical energy of a sound wave, we consider a frequency model described by the Helmholtz equation with a damping on the boundary. The well-posedness of the model is shown in a class of domains with $d$-set boundaries ($N-1\le d<N$). We introduce a...

In this article, we study the time dependent convection-diffusion-reaction equation coupled with the Darcy equation. We propose and analyze two numerical schemes based on finite element methods for the discretization in space and the implicit Euler method for the discretization in time. An optimal a priori error estimate is then derived for each nu...

The study deals with collocated Godunov type finite volume schemes applied to the two-dimensional linear wave equation with Coriolis source term. The purpose is to explain the wrong behaviour of the classic scheme and to modify it in order to avoid accuracy issues around the geostrophic equilibrium and in geostrophic adjustment processes. To do so,...

This book is the second volume of proceedings of the 8th conference on "Finite Volumes for Complex Applications" (Lille, June 2017). It includes reviewed contributions reporting successful applications in the fields of fluid dynamics, computational geosciences, structural analysis, nuclear physics, semiconductor theory and other topics.
The finite...

We present an application of the discrete duality finite volume method to the numerical approximation of the 2D Stokes or (unsteady) Navier–Stokes equations associated to Dirichlet boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete duality principles. The scheme may be seen as...

This benchmarkproposes test-cases to assess innovative finite volume type methods developped to solve the equations of incompressible fluid mechanics. Emphasis is set on the ability to handle very general meshes, on accuracy, robustness and computational complexity. Two-dimensional as well as three-dimensional tests with known analytical solutions...

The shallow water equations can be used to model many phenomena in geophysical fluid mechanics. For large scales, the Coriolis force plays an important role and the geostrophic equilibrium which corresponds to the balance between the pressure gradient and the Coriolis force is an important feature. In this communication, we investigate the stabilit...

We propose a method to explain the behaviour of the Godunov finite volume scheme applied to the linear wave equation with Coriolis source term at low Froude number. In particular, we use the Hodge decomposition and we study the properties of the modified equation associated to the Godunov scheme. Based on the structure of the discrete kernel of the...

This article is composed of three self-consistent chapters that can be read independently of each other. In Chapter 1, we define and we analyze the low Mach number problem through a linear analysis of a perturbed linear wave equation. Then, we show how to modify Godunov type schemes applied to the linear wave equation to make this scheme accurate a...

Through a linear analysis, we show how to modify Godunov type schemes applied to the compressible Euler system to make them accurate at any Mach number. This allows to propose all Mach Godunov type schemes. A linear stability result is proposed and a formal asymptotic analysis justifies the construction in the barotropic case when the Godunov type...

We derive an a posteriori error estimation for the discrete duality finite volume (DDFV) discretization of the stationary Stokes equations on very general twodimensional meshes, when a penalty term is added in the incompressibility equation to stabilize the variational formulation. Two different estimators are provided: one for the error on the vel...

We introduce continuous tools to study the low Mach number behavior of the Godunov scheme
applied to the linear wave equation with porosity on cartesian meshes. More precisely, we
extend the Hodge decomposition to a weighted L2 space in the continuous case and we
study the properties of the modified equation associated to this Godunov scheme. This...

We present an application of the discrete duality finite volume method to the numerical approximation of the vorticity-velocity-pressure formulation of the two-dimensional Stokes equations, associated to various nonstandard boundary conditions. The finite volume method is based on the use of discrete differential operators obeying some discrete dua...

We present an extension of the Optimized Schwarz Waveform Relaxation method with Robin transmission conditions to finite volume schemes of DDFV type (Discrete Duality Finite Volumes) for solving heterogeneous time-dependent advection-diffusion problems. We propose a new strategy which is well adapted to domain decomposition for coupling upwind disc...

We establish discrete Poincaré type inequalities on a two-dimensional polygonal domain covered by arbitrary, possibly nonconforming meshes. On such meshes, discrete scalar fields are defined by their values both at the cell centers and vertices, while discrete gradients are associated with the edges of the mesh, like in the discrete duality finite...

Far field simulations of underground nuclear waste disposal involve a number of challenges for numerical simulations: widely differing lengths and time-scales, highly variable coefficients and stringent accuracy requirements. In the site under consideration by the French Agency for NuclearWaste Management (ANDRA), the repository would be located in...

We investigate the accuracy of the Godunov scheme applied to the variable cross-section acoustic equations. Contrarily to the constant cross-section case, the accuracy issue of this scheme in the low Mach number regime appears even in the one-dimensional case; on the other hand, we show that it is possible to construct another Godunov type scheme w...

Cell-centered and vertex-centered finite volume schemes for the Laplace equation with homogeneous Dirichlet boundary conditions are considered on a triangular mesh and on the Voronoi diagram associated to its vertices. A broken P 1 function is constructed from the solutions of both schemes. When the domain is two-dimensional polygonal convex, it is...

By studying the structure of the discrete kernel of the linear acoustic operator discretized with a Godunov scheme, we clearly explain why the behaviour of the Godunov scheme applied to the linear wave equation deeply depends on the space dimension and, especially, on the type of mesh. This approach allows us to explain why, in the periodic case, t...

We design a Schwarz waveform relaxation algorithm for solving advection-diffusion-reaction problems in heterogeneous media. The non-overlapping domain decomposition method is global in time and thus allows the use of different time steps in different subdomains. We use a discontinuous Galerkin method in time as a subdomain solver in order to have o...

This document is a synthesis of a set of works concerning the development and the analysis of finite volume methods used for the numerical approximation of partial differential equations (PDEs) stemming from physics. In the first part, the document deals with colocalized Godunov type schemes for the Maxwell and wave equations, with a study on the l...

The cell-centered finite volume approximation of the Laplace equation in dimension one is considered. An exact expression of the error between the exact and numerical solutions is derived through the use of continuous and discrete Green functions. This allows to discuss convergence of the method in the L infinity and L2 norms with respect to the ch...

An efficient and fully computable a posteriori error bound is derived for the discrete duality finite volume discretization of the Laplace equation on very general twodimensional meshes. The main ingredients are the equivalence of this method with a finite element like scheme and tools from the finite element framework. Numerical tests are performe...

A new finite volume method is presented for discretizing the two-dimensional Maxwell equations. This method may be seen as an extension of the covolume type methods to arbitrary, possibly non-conforming or even non-convex, n-sided polygonal meshes, thanks to an appropriate choice of degrees of freedom. An equivalent formulation of the scheme is giv...

We apply the discrete duality finite volume method to the FVCA5 benchmark. The scheme runs on all tests. Second order convergence is observed in the L2 norm, while convergence in the L2 norm of the gradient is observed with an order 1 or 1.5, depending on the family of meshes. In the most severe tests, the scheme may violate the maximum principle.

The discretization of the Laplace equation by the discrete duality finite volume method is considered. The discrete variational formulation of the method and the Helmholtz-Hodge decomposition of the error enable us to split the energy norm of the error into a conforming part and a non-conforming part. Using Poincaré-type inequalities, a fully calcu...

Dans ce travail nous menons une étude a posteriori de l’erreur pour la méthode de volumes ﬁnis baptisée discrete duality ﬁnite volume (DDFV) dans le cadre de la discrétisation d'une équation de diffusion. Nous appliquons une récente technique introduite par M. Vohralík pour calculer à moindre coût des indicateurs d'erreur a posteriori complètement...

We compare two models used to compute the internal hydrodynamics of a gas centrifuge. The scoop action is taken into account through boundary conditions on the flow entering the bowl of the centrifuge in the first model, and through sinks and drag forces in the chambers of the centrifuge in the second. The numerical approximations of the models are...

We dene discrete dieren tial operators such as grad, div and curl, on general two-dimensional non-orthogonal meshes. These discrete operators verify discrete analogues of usual continuous theorems: discrete Green formulae, discrete Hodge decomposition of vector elds, vector curls have a vanishing divergence and gradients have a vanishing curl. We a...

We present a finite volume method based on the integration of the Laplace equation on both the cells of a primal almost arbitrary two-dimensional mesh and those of a dual mesh obtained by joining the centers of the cells of the primal mesh. The key ingredient is the definition of discrete gradient and divergence operators verifying a discrete Green...

This paper deals with applications of the "Discrete-Duality Finite Volume" approach to a variety of elliptic problems. This is a new finite volume method, based on the derivation of discrete operators obeying a Discrete-Duality principle. An appropriate choice of the degrees of freedom allows one to use arbitrary meshes. We show that the method is...

This paper deals with the linear response of a plasma in a one-dimensional bounded geometry under the action of a time-periodic electric field. The nonlinear Vlasov equation is solved by following the characteristic curves until they reach the boundary of the domain, where the distribution function of the incoming particles is supposed to be known...

A self-consistent nonlinear model of an isotope separation process based on selective ion cyclotron resonance heating in a magnetized plasma is presented, and its numerical resolution is described. The response of the electrons to the electromagnetic field is modeled by a cold and linear conductivity tensor, while a particle method is used to solve...

The problem of spurious solutions due to the violation of Gauss’ law in computational electromagnetics is avoided by solving an equivalent Maxwell system that takes this constraint into account. A second-order accurate finite-volume method is proposed to solve this linear, first-order strictly hyperbolic reformulated system. Numerical examples demo...

A finite-volume scheme on unstructured meshes for the three-dimensional time-dependent Maxwell equations is presented. To avoid the increase of numerical errors caused by suppressing the information contained in Gauss' law as well as the divergence-free condition of the magnetic induction, a divergence cleaning step is added which does not require...

Usually, non-stationary numerical calculations in electromagnetics are based on the hyperbolic evolution equations for the electric and magnetic fields and leave Gauss' law out of consideration because the latter is a consequence of the former and of the charge conservation equation in the continuous case. However, in the simulation of the self-con...

We present a high-resolution finite-volume Godunov-type Maxwell solver for three-dimensional unstructured meshes, based on the purely hyperbolic Maxwell system, which is established by introducing two additional degrees of freedom into the evolutionary part of the Maxwell equations and coupling them with the elliptical constraints given by Gauß’ la...

Pulsed-power diodes have been developed at the Forschungszentrum Karlsruhe and are the objects of extensive experimental as well as numerical investigations. The electrical behavior of the diodes is substantially influenced by a charged particle flow forming a non-neutral plasma inside these devices. A detailed understanding of the fundamental time...

## Projects

Project (1)