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

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

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

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

Numerical analysis of the planewave discretization of some orbital-free and Kohn-Sham models

We provide a priori error estimates for the spectral and pseudospectral Fourier (also called planewave) discretizations of the periodic Thomas-Fermi-von Weizs\"{a}cker (TFW) model and for the spectral discretization of the Kohn-Sham model, within the local density approximation (LDA). These models allow to compute approximations of the ground state energy and density of molecular systems in the condensed phase. The TFW model is stricly convex with respect to the electronic density, and allows for a comprehensive analysis. This is not the case for the Kohn-Sham LDA model, for which the uniqueness of the ground state electronic density is not guaranteed. Under a coercivity assumption on the second order optimality condition, we prove that for large enough energy cut-offs, the discretized Kohn-Sham LDA problem has a minimizer in the vicinity of any Kohn-Sham ground state, and that this minimizer is unique up to unitary transform. We then derive optimal a priori error estimates for the spectral discretization method.Comment: 50 page

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

The Microscopic Origin of the Macroscopic Dielectric Permittivity of Crystals: A Mathematical Viewpoint

The purpose of this paper is to provide a mathematical analysis of the Adler-Wiser formula relating the macroscopic relative permittivity tensor to the microscopic structure of the crystal at the atomic level. The technical level of the presentation is kept at its minimum to emphasize the mathematical structure of the results. We also briefly review some models describing the electronic structure of finite systems, focusing on density operator based formulations, as well as the Hartree model for perfect crystals or crystals with a defect.Comment: Proceedings of the Workshop "Numerical Analysis of Multiscale Computations" at Banff International Research Station, December 200

Approximating the vanishing capillarity limit of two-phase flow in multi-dimensional heterogeneous porous medium

International audienceNeglecting capillary pressure effects in two-phase flow models for porous media may lead to non-physical solutions: indeed, the physical solution is obtained as limit of the parabolic model with small but non-zero capillarity. In this paper, we propose and compare several numerical strategies designed specifically for approximating physically relevant solutions of the hyperbolic model with neglected capillarity, in the multi-dimensional case. It has been shown in [Andreianov&Canc'es, Comput. Geosci., 2013, to appear] that in the case of the one-dimensional Buckley-Leverett equation with distinct capillary pressure properties of adjacent rocks, the interface may impose an upper bound on the transmitted flux. This transmission condition may reflect the oil trapping phenomenon. We recall the theoretical results for the one-dimensional case which are used to motivate the construction of multi- dimensional finite volume schemes. We describe and compare a coupled scheme resulting as the limit of the scheme constructed in [Brenner & Canc'es & Hilhorst, HAL preprint no.00675681, 2012) and two IMplicit Pressure - Explicit Saturation (IMPES) schemes with different level of coupling