Mathematics and Finite Element Discretizations of Incompressible Navier–Stokes Flows

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 flow of incompressible Newtonian fluids 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.

This book is intended for applied mathematicians and graduate students interested in learning about the theory of various finite element methods for solving the Navier–Stokes equations. Engineers seeking reliable algorithms for computational fluid dynamics will also find the book of interest.