Aiming to provide the reader with a general overview of the mathematical and numerical techniques used for the simulation of matter at the microscopic scale, this book lays the emphasis on the numerics, but modelling aspects are also addressed. The contributors come from different scientific communities: physics, theoretical chemistry, mathematical analysis, stochastic analysis, numerical analysis, and the text should be suitable for graduate students in mathematics, sciences and engineering and technology.

This book studies the existence and uniqueness of solutions to parabolic-type equations with irregular coefficients and/or initial conditions. It elaborates on the DiPerna-Lions theory of renormalized solutions to linear transport equations and related equations, and also examines the connection between the results on the partial differential equation and the well-posedness of the underlying stochastic/ordinary differential equation.

The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering.

The thermodynamic limit is a mathematical technique for modeling crystals or other macroscopic objects by considering them as infinite periodic arrays of molecules. The technique allows models in solid state physics to be derived directly from models in quantum chemistry. This book presents new results, many previously unpublished, for a large class of models and provides a survey of the mathematics of thermodynamic limit problems. The authors both work closely with Fields Medal-winner Pierre-Louis Lion, and the book will be a valuable tool for applied mathematicians and mathematical physicists studying nonlinear partial differential equations.

This comprehensive text focuses on mathematical and numerical techniques for the simulation of magnetohydrodynamic phenomena, with an emphasis laid on the magnetohydrodynamics of liquid metals, and on a prototypical industrial application. Aimed at research mathematicians, engineers, and physicists, as well as those working in industry, and starting from a good understanding of the physics at play, the approach is a highly mathematical one, based on the rigorous analysis of the equations at hand, and a solid numerical analysis to found the simulations. At each stage of the exposition, examples of numerical simulations are provided, first on academic test cases to illustrate the approach, next on benchmarks well documented in the professional literature, and finally, whenever possible, on real industrial cases.

Systemes multi-echelles est une introduction a la problematique des systemes multi-echelles du point de vue du mathematicien applique. Il se compose d'une mosaique d'exemples dont le seul lien est d'appartenir a la tres grande famille des problemes issus de la physique au sens large qui presentent pour leur modelisation et leur simulation cette difficulte essentielle de comporter en leur sein des echelles de temps ou d'espace tres differentes.

Ce cours est une introduction A la modA(c)lisation mathA(c)matique et A l'analyse numA(c)rique pour la chimie molA(c)culaire quantique, un champ peu connu des mathA(c)maticiens et pourtant riche en sujets d'investigation. Le point de vue choisi est celui du mathA(c)maticien appliquA(c). Le cours est construit de maniA]re auto-consistante. Seules des notions de base en analyse fonctionnelle sont requises pour l'aborder. Les outils mathA(c)matiques plus A(c)laborA(c)s sont introduits progressivement et les connaissances nA(c)cessaires en physique et en thA(c)orie spectrale sont regroupA(c)es dans des annexes. On prA(c)sente d'abord les modA]les les plus utilisA(c)s en pratique. Puis, on analyse ces modA]les d'un point de vue mathA(c)matique (questions d'existence de solutions, d'unicitA(c), ...). On introduit ensuite les diffA(c)rentes stratA(c)gies numA(c)riques employA(c)es pour la rA(c)solution pratique, et on fournit, quand ceci est possible, des A(c)lA(c)ments d'analyse numA(c)rique de ces mA(c)thodes. Les liens existants entre les modA]les de la chimie molA(c)culaire et des sujets connexes sont aussi explorA(c)s: modA(c)lisation de la phase liquide, physique de l'A(c)tat cristallin, biologie, simulation des matA(c)riaux, ... Le cours peut aussi intA(c)resser le chimiste ou le physicien curieux de comprendre les techniques mathA(c)matiques dont relA]vent les modA]les qu'il utilise, et de dA(c)couvrir comment de telles techniques peuvent amA(c)liorer significativement l'efficacitA(c) et la qualitA(c) des simulations numA(c)riques.

The book is an introduction to the mathematical modelling and the numerical analysis for computational quantum chemistry. The text isself-consistent: no particular familiarity with the subject nor the mathematical tools is required. Only a basic training in functional analysis is necessary. The models are introduced and discussed. They are analyzed mathematically. The numerical techniques are presented and, when possible, also analyzed. Topics closely related to molecular simulation are also addressed, such as solid state physics, liquid state physics, biology, and materials science. The book can
also be useful for students and researchers of those fields, curious about the mathematical techniques and the numerical strategies.

The book provides a pedagogic and comprehensive introduction to homogenization theory with a special focus on problems set for non-periodic media. The presentation encompasses both deterministic and probabilistic settings. It also mixes the most abstract aspects with some more practical aspects regarding the numerical approaches necessary to simulate such multiscale problems. Based on lecture courses of the authors, the book is suitable for graduate students of mathematics and engineering. Cet ouvrage est une introduction pedagogique a la theorie de l'homogeneisation, et aux approches numeriques associees, pour la comprehension et la simulation des problemes a plusieurs echelles. La presentation est axee sur les differentes hypotheses possibles pour mettre en oeuvre la theorie, selon que le milieu ambiant a une geometrie periodique ou non. Le public vise est celui des cycles M et D des universites ainsi que celui des ecoles d'ingenieurs et formations equivalentes.