This book presents a comprehensive treatment of the mathematical developments in the finite element approximation of the Navier-Stokes equations. It focuses on mixed finite element methods, which are fundamental in numerical hydrodynamics. The text is self-contained, with a review of theoretical results on the Navier-Stokes problem and references to R. Temam's book. The authors also acknowledge M. Crouzeix for his contributions. The book is divided into five chapters. Chapter I provides the mathematical foundation of the Stokes problem, covering elliptic boundary value problems, function spaces, vector field decompositions, and the analysis of an abstract variational problem. Chapter II discusses the numerical solution of the Stokes problem using a classical method, including approximation results and examples of first- and second-order approximations. Chapter III presents a mixed finite element method for solving the Stokes problem, including a mixed formulation and application to finite elements of degree 2. Chapter IV addresses the stationary Navier-Stokes equations, discussing nonlinear problems, solutions, and mixed methods for approximation. Chapter V deals with the time-dependent Navier-Stokes equations, including the continuous problem, semi-discretization methods, and convergence of multistep methods. The book is intended as a graduate-level course in numerical analysis and serves as an introduction to mixed finite element theory. It includes bibliographical notes, references, an index, and an appendix. The work is published by Springer-Verlag and was originally taught at the University Pierre & Marie Curie.This book presents a comprehensive treatment of the mathematical developments in the finite element approximation of the Navier-Stokes equations. It focuses on mixed finite element methods, which are fundamental in numerical hydrodynamics. The text is self-contained, with a review of theoretical results on the Navier-Stokes problem and references to R. Temam's book. The authors also acknowledge M. Crouzeix for his contributions. The book is divided into five chapters. Chapter I provides the mathematical foundation of the Stokes problem, covering elliptic boundary value problems, function spaces, vector field decompositions, and the analysis of an abstract variational problem. Chapter II discusses the numerical solution of the Stokes problem using a classical method, including approximation results and examples of first- and second-order approximations. Chapter III presents a mixed finite element method for solving the Stokes problem, including a mixed formulation and application to finite elements of degree 2. Chapter IV addresses the stationary Navier-Stokes equations, discussing nonlinear problems, solutions, and mixed methods for approximation. Chapter V deals with the time-dependent Navier-Stokes equations, including the continuous problem, semi-discretization methods, and convergence of multistep methods. The book is intended as a graduate-level course in numerical analysis and serves as an introduction to mixed finite element theory. It includes bibliographical notes, references, an index, and an appendix. The work is published by Springer-Verlag and was originally taught at the University Pierre & Marie Curie.