Alexandre Ern - Böcker

The origins of the finite element method can be traced back to the 1950s when engineers started to solve numerically structural mechanics problems in aeronautics. Since then, the field of applications has widened steadily and nowadays encompasses nonlinear solid mechanics, fluid/structure interactions, flows in industrial or geophysical settings, multicomponent reactive turbulent flows, mass transfer in porous media, viscoelastic flows in medical sciences, electromagnetism, wave scattering problems, and option pricing (to cite a few examples). Numerous commercial and academic codes based on the finite element method have been developed over the years. The method has been so successful to solve Partial Differential Equations (PDEs) that the term "Finite Element Method" nowadays refers not only to the mere interpolation technique it is, but also to a fuzzy set of PDEs and approximation techniques. The efficiency of the finite element method relies on two distinct ingredi­ ents: the interpolation capability of finite elements (referred to as the approx­ imability property in this book) and the ability of the user to approximate his model (mostly a set of PDEs) in a proper mathematical setting (thus guar­ anteeing continuity, stability, and consistency properties). Experience shows that failure to produce an approximate solution with an acceptable accuracy is almost invariably linked to departure from the mathematical foundations. Typical examples include non-physical oscillations, spurious modes, and lock­ ing effects. In most cases, a remedy can be designed if the mathematical framework is properly set up.

The origins of the finite element method can be traced back to the 1950s when engineers started to solve numerically structural mechanics problems in aeronautics. Since then, the field of applications has widened steadily and nowadays encompasses nonlinear solid mechanics, fluid/structure interactions, flows in industrial or geophysical settings, multicomponent reactive turbulent flows, mass transfer in porous media, viscoelastic flows in medical sciences, electromagnetism, wave scattering problems, and option pricing (to cite a few examples). Numerous commercial and academic codes based on the finite element method have been developed over the years. The method has been so successful to solve Partial Differential Equations (PDEs) that the term "Finite Element Method" nowadays refers not only to the mere interpolation technique it is, but also to a fuzzy set of PDEs and approximation techniques. The efficiency of the finite element method relies on two distinct ingredi­ ents: the interpolation capability of finite elements (referred to as the approx­ imability property in this book) and the ability of the user to approximate his model (mostly a set of PDEs) in a proper mathematical setting (thus guar­ anteeing continuity, stability, and consistency properties). Experience shows that failure to produce an approximate solution with an acceptable accuracy is almost invariably linked to departure from the mathematical foundations. Typical examples include non-physical oscillations, spurious modes, and lock­ ing effects. In most cases, a remedy can be designed if the mathematical framework is properly set up.

This volume gathers contributions from participants of the Introductory School and the IHP thematic quarter on Numerical Methods for PDE, held in 2016 in Cargese (Corsica) and Paris, providing an opportunity to disseminate the latest results and envisage fresh challenges in traditional and new application fields. Numerical analysis applied to the approximate solution of PDEs is a key discipline in applied mathematics, and over the last few years, several new paradigms have appeared, leading to entire new families of discretization methods and solution algorithms. This book is intended for researchers in the field.

To engage the reader, the text combines examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy.Volume I is divided into 23 chapters plus two appendices on Banach and Hilbert spaces and on differential calculus.

To engage the reader, the text combines examples, basic ideas, rigorous proofs, and pointers to the literature to enhance scientific literacy.Volume I is divided into 23 chapters plus two appendices on Banach and Hilbert spaces and on differential calculus.

These techniques are applied to elliptic PDEs (diffusion, elasticity, the Helmholtz problem, Maxwell's equations), eigenvalue problems for elliptic PDEs, and PDEs in mixed form (Darcy and Stokes flows).

These techniques are applied to elliptic PDEs (diffusion, elasticity, the Helmholtz problem, Maxwell's equations), eigenvalue problems for elliptic PDEs, and PDEs in mixed form (Darcy and Stokes flows).

The remainder of Volume III addresses time-dependent problems: parabolic equations (such as the heat equation), evolution equations without coercivity (Stokes flows, Friedrichs' systems), and nonlinear hyperbolic equations (scalar conservation equations, hyperbolic systems).

The remainder of Volume III addresses time-dependent problems: parabolic equations (such as the heat equation), evolution equations without coercivity (Stokes flows, Friedrichs' systems), and nonlinear hyperbolic equations (scalar conservation equations, hyperbolic systems).

The book is primarily intended for graduate students, researchers, and engineers working in related fields of application, and it can also be used as a support for graduate and doctoral lectures.

This volume gathers contributions from participants of the Introductory School and the IHP thematic quarter on Numerical Methods for PDE, held in 2016 in Cargese (Corsica) and Paris, providing an opportunity to disseminate the latest results and envisage fresh challenges in traditional and new application fields. Numerical analysis applied to the approximate solution of PDEs is a key discipline in applied mathematics, and over the last few years, several new paradigms have appeared, leading to entire new families of discretization methods and solution algorithms. This book is intended for researchers in the field.

Ces notes de cours (Ecole Nationale des Ponts et Chaussées, DEA de Mécanique de Paris VI) présentent la méthode des éléments finis dans un cadre mathématique rigoureux. En accordant une place fondamentale aux conditions inf-sup, elles s'affranchissent du cadre réducteur Lax-Milgram/Galerkin standard. Elles couvrent un spectre d'applications relativement large et apportent de nombreuses précisions sur la mise en oeuvre numérique. Trois plans de lecture sont proposés: le premier conçu pour un lecteur intéressé par les aspects mathématiques, le deuxième s' adressant aux ingénieurs et le troisième limité aux aspects élémentaires. Les prérequis mathématiques, de niveau 2ème cycle universitaire, sont rappelés dans deux annexes.

This book introduces the basic ideas to build discontinuous Galerkin methods and, at the same time, incorporates several recent mathematical developments. The presentation is to a large extent self-contained and is intended for graduate students and researchers in numerical analysis. The material covers a wide range of model problems, both steady and unsteady, elaborating from advection-reaction and diffusion problems up to the Navier-Stokes equations and Friedrichs' systems. Both finite element and finite volume viewpoints are exploited to convey the main ideas underlying the design of the approximation. The analysis is presented in a rigorous mathematical setting where discrete counterparts of the key properties of the continuous problem are identified. The framework encompasses fairly general meshes regarding element shapes and hanging nodes. Salient implementation issues are also addressed.

With the advent of sophisticated computer technology and the development of efficient computational algorithms, numerical modeling of complex multicomponent laminar reacting flows has emerged as an increasingly popular and firmly established area of scientific research. Progress in this area aims at obtaining better resolved and more accurate solutions of specific technological problems in less computer time. Therefore, it strongly relies upon the ability of evaluating fundamental parameters appearing in the physical models. Transport properties constitute a typical example of the above characterization. Evaluating transport coefficients of dilute polyatomic gas mixtures is often critical in many engineering applications, including chemical reactors, hypersonic flows, comb- tion phenomena, and chemical vapor deposition. Using the kinetic theory of dilute polyatomic gas mixtures as a starting point, this book offers a systematic development of a mathematical and numerical theory for the evaluation of transport properties in dilute polyatomic gas mixtures. The present investigation is not specifically.about the kinetic theory of gases, for which there are plenty of excellent and thoroughly do- mented textbooks; it is rather geared toward the development of new, efficient, and general algorithms with which to evaluate transport properties of dilute polyatomic gas mixtures at a reasonable computational cost.