Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution

We consider an immiscible two-phase flow in a heterogeneous one-dimensional porous medium. We suppose particularly that the capillary pressure field is discontinuous with respect to the space variable. The dependence of the capillary pressure with respect to the oil saturation is supposed to be weak, at least for saturations which are not too close to 0 or 1. We study the asymptotic behavior when the capillary pressure tends to a function which does not depend on the saturation. In this paper, we show that if the capillary forces at the spacial discontinuities are oriented in the same direction that the gravity forces, or if the two phases move in the same direction, then the saturation profile with capillary diffusion converges toward the unique optimal entropy solution to the hyperbolic scalar conservation law with discontinuous flux functions

Improving Newton's method performance by parametrization: the case of Richards equation

The nonlinear systems obtained by discretizing degenerate parabolic equations may be hard to solve, especially with Newton's method. In this paper, we apply to Richards equation a strategy that consists in defining a new primary unknown for the continuous equation in order to stabilize Newton's method by parametrizing the graph linking the pressure and the saturation. The resulting form of Richards equation is then discretized thanks to a monotone Finite Volume scheme. We prove the well-posedness of the numerical scheme. Then we show under appropriate non-degeneracy conditions on the parametrization that Newton\^as method converges locally and quadratically. Finally, we provide numerical evidences of the efficiency of our approach

On the time continuity of entropy solutions

We show that any entropy solution $u$ of a convection diffusion equation $\partial_t u + \div F(u)-\Delta\phi(u) =b$ in \OT belongs to C([0,T),L^1_{Loc}(\o\O)). The proof does not use the uniqueness of the solution

Numerical analysis of a robust free energy diminishing Finite Volume scheme for parabolic equations with gradient structure

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach

Local defects are always neutral in the Thomas-Fermi-von Weisz\"acker theory of crystals

The aim of this article is to propose a mathematical model describing the electronic structure of crystals with local defects in the framework of the Thomas-Fermi-von Weizs\"acker (TFW) theory. The approach follows the same lines as that used in {\it E. Canc\`es, A. Deleurence and M. Lewin, Commun. Math. Phys., 281 (2008), pp. 129--177} for the reduced Hartree-Fock model, and is based on thermodynamic limit arguments. We prove in particular that it is not possible to model charged defects within the TFW theory of crystals. We finally derive some additional properties of the TFW ground state electronic density of a crystal with a local defect, in the special case when the host crystal is modelled by a homogeneous medium.Comment: 34 page

Incompressible immiscible multiphase flows in porous media: a variational approach

We describe the competitive motion of (N + 1) incompressible immiscible phases within a porous medium as the gradient flow of a singular energy in the space of non-negative measures with prescribed mass endowed with some tensorial Wasserstein distance. We show the convergence of the approximation obtained by a minimization schem\`e a la [R. Jordan, D. Kinder-lehrer \& F. Otto, SIAM J. Math. Anal, 29(1):1--17, 1998]. This allow to obtain a new existence result for a physically well-established system of PDEs consisting in the Darcy-Muskat law for each phase, N capillary pressure relations, and a constraint on the volume occupied by the fluid. Our study does not require the introduction of any global or complementary pressure

A mathematical analysis of the GW0 method for computing electronic excited energies of molecules

This paper analyses the GW method for finite electronic systems. In a first step, we provide a mathematical framework for the usual one-body operators that appear naturally in many-body perturbation theory. We then discuss the GW equations which construct an approximation of the one-body Green's function, and give a rigorous mathematical formulation of these equations. Finally, we study the well-posedness of the GW0 equations, proving the existence of a unique solution to these equations in a perturbative regime

Greedy algorithms for high-dimensional eigenvalue problems

In this article, we present two new greedy algorithms for the computation of the lowest eigenvalue (and an associated eigenvector) of a high-dimensional eigenvalue problem, and prove some convergence results for these algorithms and their orthogonalized versions. The performance of our algorithms is illustrated on numerical test cases (including the computation of the buckling modes of a microstructured plate), and compared with that of another greedy algorithm for eigenvalue problems introduced by Ammar and Chinesta.Comment: 33 pages, 5 figure