Ce livre s'adresse aux etudiants qui souhaitent apprendre les bases de l'analyse numerique des equations aux derivees partielles (EDPs) sans se heurter a un formalisme mathematique trop abstrait.Il presente, de la maniere la plus elementaire et la plus pedagogique possible, la demarche complete du mathematicien applique s'attaquant a une EDP issue d'un modele physique, biologique ou economique : d'abord l'etude theorique de l'equation, ensuite la determination de schemas numeriques pertinents pour calculer une bonne approximation de la solution et enfin la resolution pratique de ces schemas par des algorithmes adaptes.Cette demarche est illustree sur des exemples tres simples mais le lecteur doit se confronter a des questions mathematiques variees : existence et unicite des solutions d'EDPs, mise en uvre de schemas numeriques, methodes de resolution de systemes lineaires ou de problemes d'optimisation, etc. Il est aussi invite a mettre en pratique ses connaissances via de nombreux exercices et des problemes de revision.La partie differences finies est accessible a un etudiant d'un niveau equivalent a une Licence de Mathematiques. La partie elements finis est destinee aux etudiants de Master 2 de Mathematiques ou preparant l'Agregation car elle necessite des resultats d'analyse fonctionnelle.

This monograph presents the most recent developments in the study of Hamilton-Jacobi Equations and control problems with discontinuities, mainly from the viewpoint of partial differential equations. Two main cases are investigated in detail: the case of codimension 1 discontinuities and the stratified case in which the discontinuities can be of any codimensions. In both, connections with deterministic control problems are carefully studied, and numerous examples and applications are illustrated throughout the text.After an initial section that provides a “toolbox” containing key results which will be used throughout the text, Parts II and III completely describe several recently introduced approaches to treat problems involving either codimension 1 discontinuities or networks. The remaining sections are concerned with stratified problems either in the whole space R^N or in bounded or unbounded domains with state-constraints. In particular, the use of stratified solutions to treat problems with boundary conditions, where both the boundary may be non-smooth and the data may present discontinuities, is developed. Many applications to concrete problems are explored throughout the text – such as Kolmogorov-Petrovsky-Piskunov (KPP) type problems, large deviations, level-sets approach, large time behavior, and homogenization – and several key open problems are presented.This monograph will be of interest to graduate students and researchers working in deterministic control problems and Hamilton-Jacobi Equations, network problems, or scalar conservation laws.

This monograph presents the most recent developments in the study of Hamilton-Jacobi Equations and control problems with discontinuities, mainly from the viewpoint of partial differential equations. Two main cases are investigated in detail: the case of codimension 1 discontinuities and the stratified case in which the discontinuities can be of any codimensions. In both, connections with deterministic control problems are carefully studied, and numerous examples and applications are illustrated throughout the text.After an initial section that provides a “toolbox” containing key results which will be used throughout the text, Parts II and III completely describe several recently introduced approaches to treat problems involving either codimension 1 discontinuities or networks. The remaining sections are concerned with stratified problems either in the whole space R^N or in bounded or unbounded domains with state-constraints. In particular, the use of stratified solutions to treat problems with boundary conditions, where both the boundary may be non-smooth and the data may present discontinuities, is developed. Many applications to concrete problems are explored throughout the text – such as Kolmogorov-Petrovsky-Piskunov (KPP) type problems, large deviations, level-sets approach, large time behavior, and homogenization – and several key open problems are presented.This monograph will be of interest to graduate students and researchers working in deterministic control problems and Hamilton-Jacobi Equations, network problems, or scalar conservation laws.

Ce livre s'adresse aux etudiants qui souhaitent apprendre les bases de l'analyse numerique des equations aux derivees partielles (EDPs) sans se heurter a un formalisme mathematique trop abstrait.Il presente, de la maniere la plus elementaire et la plus pedagogique possible, la demarche complete du mathematicien applique s'attaquant a une EDP issue d'un modele physique, biologique ou economique : d'abord l'etude theorique de l'equation, ensuite la determination de schemas numeriques pertinents pour calculer une bonne approximation de la solution et enfin la resolution pratique de ces schemas par des algorithmes adaptes.Cette demarche est illustree sur des exemples tres simples mais le lecteur doit se confronter a des questions mathematiques variees : existence et unicite des solutions d'EDPs, mise en uvre de schemas numeriques, methodes de resolution de systemes lineaires ou de problemes d'optimisation, etc. Il est aussi invite a mettre en pratique ses connaissances via de nombreux exercices et des problemes de revision.La partie differences finies est accessible a un etudiant d'un niveau equivalent a une Licence de Mathematiques. La partie elements finis est destinee aux etudiants de Master 2 de Mathematiques ou preparant l'Agregation car elle necessite des resultats d'analyse fonctionnelle.

This monograph presents the most recent developments in the study of Hamilton-Jacobi Equations and control problems with discontinuities, mainly from the viewpoint of partial differential equations. Two main cases are investigated in detail: the case of codimension 1 discontinuities and the stratified case in which the discontinuities can be of any codimensions. In both, connections with deterministic control problems are carefully studied, and numerous examples and applications are illustrated throughout the text.After an initial section that provides a “toolbox” containing key results which will be used throughout the text, Parts II and III completely describe several recently introduced approaches to treat problems involving either codimension 1 discontinuities or networks. The remaining sections are concerned with stratified problems either in the whole space R^N or in bounded or unbounded domains with state-constraints. In particular, the use of stratified solutions to treat problems with boundary conditions, where both the boundary may be non-smooth and the data may present discontinuities, is developed. Many applications to concrete problems are explored throughout the text – such as Kolmogorov-Petrovsky-Piskunov (KPP) type problems, large deviations, level-sets approach, large time behavior, and homogenization – and several key open problems are presented.This monograph will be of interest to graduate students and researchers working in deterministic control problems and Hamilton-Jacobi Equations, network problems, or scalar conservation laws.

This monograph presents the most recent developments in the study of Hamilton-Jacobi Equations and control problems with discontinuities, mainly from the viewpoint of partial differential equations. Two main cases are investigated in detail: the case of codimension 1 discontinuities and the stratified case in which the discontinuities can be of any codimensions. In both, connections with deterministic control problems are carefully studied, and numerous examples and applications are illustrated throughout the text.After an initial section that provides a “toolbox” containing key results which will be used throughout the text, Parts II and III completely describe several recently introduced approaches to treat problems involving either codimension 1 discontinuities or networks. The remaining sections are concerned with stratified problems either in the whole space R^N or in bounded or unbounded domains with state-constraints. In particular, the use of stratified solutions to treat problems with boundary conditions, where both the boundary may be non-smooth and the data may present discontinuities, is developed. Many applications to concrete problems are explored throughout the text – such as Kolmogorov-Petrovsky-Piskunov (KPP) type problems, large deviations, level-sets approach, large time behavior, and homogenization – and several key open problems are presented.This monograph will be of interest to graduate students and researchers working in deterministic control problems and Hamilton-Jacobi Equations, network problems, or scalar conservation laws.