Navier-Stokes equations are one of the most impactful techniques for modeling physical flow phenomena. The coupling of velocity and pressure, along with the nonlinearity, is a challenge for the mathematical and numerical analysis of these equations. This self-contained book provides a thorough theoretical study of finite element methods for solving incompressible Navier-Stokes equations, which model ?ow of incompressible Newtonian ?uids and are used in many practical applications. It focuses on efficient and widely used finite element methods that are well adapted to large-scale simulations. In this revised and expanded edition of Girault and Raviart's 1986 textbook Finite Element Methods for Navier-Stokes Equations (Springer-Verlag), readers will find rigorous proof of stability and convergence, analysis of practical algorithms, and a stand-alone chapter on finite element methods that is applicable to a large range of PDEs. In addition to the basic theoretical analysis, this book covers up-to-date finite element discretizations of incompressible Navier-Stokes equations; a variety of numerical algorithms used in the computer implementation of Navier-Stokes equations and numerical experiments; standard and nonstandard boundary conditions and their numerical discretizations via the finite element methods; and conforming and nonconforming finite elements, as well as their stability and instability.
Christine Bernardi was a senior researcher at CNRS in the Laboratoire Jacques-Louis Lions at Sorbonne University. She received the Blaise Pascal Prize in 1995 for her outstanding research in numerical analysis, coauthored three books, and published more than 100 scientific publications before her death in 2018. Vivette Girault is an Emeritus Professor in the Laboratoire Jacques-Louis Lions at Sorbonne University. She received the AWM-SIAM Sonia Kovalevsky Lecturer award in 2021. She has authored or coauthored hundreds of scientific publications and coauthored three books on finite element methods and on non-Newtonian fluids. Frederic Hecht is an Emeritus Professor in the Laboratoire Jacques-Louis Lions at Sorbonne University. An expert in scientific computing, he has authored or coauthored more than 200 scientific publications and two books. He is the main developer of the software FreeFEM and received the IBM Prize "Intensive Numerical Computing of Automatic Generation of Three-Dimensional Meshes" in 1989, the Fondation d'Entreprise EADS (Sciences et Ingenierie) Prize in 2013, and the Atos-Joseph Fourier prize (formerly known as Bull Joseph Fourier) in 2015. Pierre-Arnaud Raviart is an Emeritus Professor in the Laboratoire Jacques-Louis Lions at Sorbonne University. He is an expert in partial differential equations and numerical methods in fluid mechanics and has authored or coauthored more than 100 scientific publications and eight books. He founded with P.G. Ciarlet the theoretical analysis of finite element methods. He is a member of the French Academy of Sciences. Beatrice Riviere is a Noah Harding Chair and Professor at Rice University in the department of Computational Applied Mathematics and Operations Research. She is an expert in the theory and implementation of numerical methods for the solution of PDEs arising from porous media and computational fluid dynamics and has authored or coauthored more than 100 scientific publications as well as a book on discontinuous Galerkin methods (SIAM, 2008). She is a SIAM Fellow (2021), an AWM Fellow (2022), and an IACM Fellow (2024).
