2022 | Book

Finite Volume Methods for the Incompressible Navier–Stokes Equations

Authors: Prof. Dr. Jian Li, Prof. Dr. Xiaolin Lin, Prof. Dr. Zhangxing Chen

Publisher: Springer International Publishing

Book Series : SpringerBriefs in Applied Sciences and Technology

Part of: Springer Professional "Business + Economics & Engineering + Technology" , Springer Professional "Engineering + Technology" , Springer Professional "Business + Economics"

Table of Contents

About this book

The book aims to provide a comprehensive understanding of the most recent developments in finite volume methods. Its focus is on the development and analysis of these methods for the two- and three-dimensional Navier-Stokes equations, supported by extensive numerical results. It covers the most used lower-order finite element pairs, with well-posedness and optimal analysis for these finite volume methods.The authors have attempted to make this book self-contained by offering complete proofs and theoretical results. While most of the material presented has been taught by the authors in a number of institutions over the past several years, they also include several updated theoretical results for the finite volume methods for the incompressible Navier-Stokes equations. This book is primarily developed to address research needs for students and academic and industrial researchers. It is particularly valuable as a research reference in the fields of engineering, mathematics, physics, and computer sciences.

Frontmatter

Chapter 1. Mathematical Foundation

Abstract

The purpose of this chapter is to recall the basic concepts related to the Navier–Stokes (NS, in short) equations, including the Sobolev spaces, several important inequalities for analysis, and the NS equations. For more details, we refer the reader to the monographs [1, 59, 122] and references therein.

Jian Li, Xiaolin Lin, Zhangxing Chen

Chapter 2. FVMs for the Stationary Stokes Equations

Abstract

In this chapter, we mainly present the construction and analysis of the stabilized lower order finite volume methods (FVMs) and adaptive finite volume methods for the stationary Stokes equations. An optimal convergence rate for stabilization of lower order finite volume methods is obtained with the same order as that of the corresponding finite element methods. In addition, after proving a number of technical lemmas (such as weighted \(L^2\)-norm estimates for regularized Green’s functions associated with the Stokes problem), optimal error estimates for the finite volume methods in the \(L^\infty \)-norm are derived for velocity gradient and pressure without a logarithmic factor \(O(|\log h|)\) for the stationary Stokes equations. Finally, convergence of the adaptive stabilized mixed finite volume methods is discussed.

Jian Li, Xiaolin Lin, Zhangxing Chen

Chapter 3. FVMs for the Stationary Navier–Stokes Equations

Abstract

In this chapter, we mainly introduce the stabilized lower order finite volume methods for the stationary Navier–Stokes equations. Due to both the practical and theoretical needs, we will consider both small data and large data for the construction of the finite volume methods of the stationary Navier–Stokes equations.

Jian Li, Xiaolin Lin, Zhangxing Chen

Chapter 4. FVMs for the Nonstationary Navier–Stokes Equations

Abstract

In this chapter, we mainly discuss the stabilized lower order finite volume methods for the nonstationary Navier–Stokes equations. Under the same regularity assumption on the exact solution for the corresponding stabilized finite element methods, the stabilized finite volume methods provide approximate solutions with the same optimal convergence rates for \(u_h\) in the \(H^1\) norm and \(p_h\) in the \(L^2\) norm. Furthermore, under a slightly different regularity assumption on the source term, the stabilized finite volume methods can also reach the optimal convergence rate for \(u_h\) in the \(L^2\) norm.

Jian Li, Xiaolin Lin, Zhangxing Chen

Backmatter

Title: Finite Volume Methods for the Incompressible Navier–Stokes Equations
Authors: Prof. Dr. Jian Li
Prof. Dr. Xiaolin Lin
Prof. Dr. Zhangxing Chen
Publisher: Springer International Publishing
Electronic ISBN: 978-3-030-94636-4
Print ISBN: 978-3-030-94635-7
DOI: https://doi.org/10.1007/978-3-030-94636-4