This publication, titled "Finite Element Approximation of the Navier-Stokes Equations," is a revised reprint of the first edition and is part of the Lecture Notes in Mathematics series edited by A. Dold and B. Eckmann. The authors, Vivette Girault and Pierre-Arnaud Raviart, are affiliated with the University Pierre et Marie Curie in Paris, France. The book provides a comprehensive treatment of the mathematical developments in the numerical solution of the Navier-Stokes equations using finite element methods. It emphasizes mixed finite element methods, which are crucial in numerical hydrodynamics. The content is designed to be self-contained, covering theoretical results and recent developments in the field. The book is structured into five chapters: 1. **Mathematical Foundation of the Stokes Problem**: This chapter covers the basic concepts of Sobolev spaces, abstract elliptic theory, and the analysis of an abstract variational problem. It also includes a decomposition of vector fields and the theory of the Stokes problem. 2. **Numerical Solution of the Stokes Problem: A Classical Method**: This chapter discusses an abstract approximation result and introduces a first method for solving the Stokes problem, including examples of first- and second-order approximations. 3. **A Mixed Finite Element Method for Solving the Stokes Problem**: This chapter focuses on mixed finite element methods, including a mixed variational problem and its application to the homogeneous Stokes problem. 4. **The Stationary Navier-Stokes Equations**: This chapter addresses a class of nonlinear problems and their application to the Navier-Stokes equations. It covers functional analysis results, solutions, and methods for approximating the Navier-Stokes equations, including a mixed method. 5. **The Time-Dependent Navier-Stokes Equations**: This chapter discusses the continuous problem, numerical solutions by semi-discretization, and multistep methods for solving the Navier-Stokes equations, including convergence results. The book includes bibliographical notes, references, an index, and an appendix. It is intended for graduate students and researchers in numerical analysis and fluid dynamics.This publication, titled "Finite Element Approximation of the Navier-Stokes Equations," is a revised reprint of the first edition and is part of the Lecture Notes in Mathematics series edited by A. Dold and B. Eckmann. The authors, Vivette Girault and Pierre-Arnaud Raviart, are affiliated with the University Pierre et Marie Curie in Paris, France. The book provides a comprehensive treatment of the mathematical developments in the numerical solution of the Navier-Stokes equations using finite element methods. It emphasizes mixed finite element methods, which are crucial in numerical hydrodynamics. The content is designed to be self-contained, covering theoretical results and recent developments in the field. The book is structured into five chapters: 1. **Mathematical Foundation of the Stokes Problem**: This chapter covers the basic concepts of Sobolev spaces, abstract elliptic theory, and the analysis of an abstract variational problem. It also includes a decomposition of vector fields and the theory of the Stokes problem. 2. **Numerical Solution of the Stokes Problem: A Classical Method**: This chapter discusses an abstract approximation result and introduces a first method for solving the Stokes problem, including examples of first- and second-order approximations. 3. **A Mixed Finite Element Method for Solving the Stokes Problem**: This chapter focuses on mixed finite element methods, including a mixed variational problem and its application to the homogeneous Stokes problem. 4. **The Stationary Navier-Stokes Equations**: This chapter addresses a class of nonlinear problems and their application to the Navier-Stokes equations. It covers functional analysis results, solutions, and methods for approximating the Navier-Stokes equations, including a mixed method. 5. **The Time-Dependent Navier-Stokes Equations**: This chapter discusses the continuous problem, numerical solutions by semi-discretization, and multistep methods for solving the Navier-Stokes equations, including convergence results. The book includes bibliographical notes, references, an index, and an appendix. It is intended for graduate students and researchers in numerical analysis and fluid dynamics.