Finite Volume Methods for the Incompressible Navier–Stokes Equations (SpringerBriefs in Applied Sciences and Technology)

✍ Scribed by Jian Li

• Publisher: Springer
• Year: 2022
• Tongue: English
• Leaves: 129
• Edition: 1st ed. 2022
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Preface
Contents
1 Mathematical Foundation
1.1 Lp Spaces
1.2 Sobolev Spaces
1.3 Imbedding Inequalities
1.4 The Navier–Stokes Equations
1.4.1 Mathematical Model
1.4.2 Numerical Solution
2 FVMs for the Stationary Stokes Equations
2.1 Introduction
2.2 Weak Formulation
2.3 Galerkin FV Approximation
2.4 Existence and Uniqueness Theorems
2.5 Priori Estimate
2.5.1 Superclose
2.5.2 Optimal Analysis
2.5.3 L2 Estimate for Velocity
2.5.4 Optimal Linfty Estimate
2.6 Posteriori Estimation
2.6.1 Upper Bound
2.6.2 Lower Bound
2.7 Adaptive Mixed Finite Volume Methods
2.7.1 Discrete Local Lower Bound
2.7.2 Adaptive Finite Volume Algorithms
2.7.3 Convergence Analysis
2.8 Numerical Experiments
2.9 Conclusions
3 FVMs for the Stationary Navier–Stokes Equations
3.1 Introduction
3.2 FVMs for the Stationary Navier–Stokes with Small Data
3.2.1 The Weak Formulation
3.2.2 Galerkin FV Approximation
3.2.3 Existence and Uniqueness Theorem
3.2.4 Convergence Analysis
3.2.5 Optimal Linfty Analysis
3.3 FVEs of Branches of Nonsingular Solutions
3.3.1 An Abstract Framework
3.3.2 Existence and Uniqueness
3.3.3 Optimal Analysis
3.3.4 Optimal Linfty Estimate
3.4 Numerical Experiments
3.5 Conclusions
4 FVMs for the Nonstationary Navier–Stokes Equations
4.1 Introduction
4.2 The Weak Formulation
4.3 Galerkin FV Approximation
4.4 Stability and Error Analysis
4.5 L2-Error Estimates
4.6 Conclusions
Appendix References
Index