This graduate-level text provides an application oriented introduction to the numerical methods for elliptic and parabolic partial differential equations. It covers finite difference, finite element, and finite volume methods, interweaving theory and applications throughout. The book examines modern topics such as adaptive methods, multilevel methods, and methods for convection-dominated problems and includes detailed illustrations and extensive exercises. For students with mathematics major it is an excellent introduction to the theory and methods, guiding them in the selection of methods and helping them to understand and pursue finite element programming. For engineering and physics students it provides a general framework for the formulation and analysis of methods.
This second edition sees additional chapters on mixed discretization and on generalizing and unifying known approaches; broader applications on systems of diffusion, convection and reaction; enhanced chapters on node-centered finite volume methods and methods of convection-dominated problems, specifically treating the now-popular cell-centered finite volume method; and the consideration of realistic formulations beyond the Poisson's equation for all models and methods.
Author(s): Peter Knabner, Lutz Angermann
Series: Texts in Applied Mathematics 44
Edition: 2
Publisher: Springer Nature Switzerland AG
Year: 2021
Language: English
Pages: 802
City: Cham
Tags: Numerical Analysis, Finite Difference Method, Finite Element Method, Finite Volume Method
Preface to the Second English Edition
From the Preface to the First English Edition
Preface to the German Edition
Contents
0 For Example: Modelling Processes in Porous Media with Differential Equations
0.1 The Basic Partial Differential Equation Models
0.2 Reactions and Transport in Porous Media
0.3 Fluid Flow in Porous Media
0.4 Reactive Solute Transport in Porous Media
0.5 Boundary and Initial Value Problems
1 For the Beginning: The Finite Difference Method for the Poisson Equation
1.1 The Dirichlet Problem for the Poisson Equation
1.2 The Finite Difference Method
1.3 Generalizations and Limitations of the Finite Difference Method
1.4 Maximum Principles and Stability
2 The Finite Element Method for the Poisson Equation
2.1 Variational Formulation for the Model Problem
2.2 The Finite Element Method with Linear Elements
2.3 Stability and Convergence of the Finite Element Method
2.4 The Implementation of the Finite Element Method: Part 1
2.5 Solving Sparse Systems of Linear Equations by Direct Methods
3 The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order
3.1 Variational Equations and Sobolev Spaces
3.2 Elliptic Boundary Value Problems of Second Order
3.3 Element Types and Affine Equivalent Partitions
3.4 Convergence Rate Estimates
3.5 The Implementation of the Finite Element Method: Part 2
3.6 Convergence Rate Results in the Case of Quadrature and Interpolation
3.7 The Condition Number of Finite Element Matrices
3.8 General Domains and Isoparametric Elements
3.9 The Maximum Principle for Finite Element Methods
4 Grid Generation and A Posteriori Error Estimation
4.1 Grid Generation
4.2 A Posteriori Error Estimates
4.3 Convergence of Adaptive Methods
5 Iterative Methods for Systems of Linear Equations
5.1 Linear Stationary Iterative Methods
5.2 Gradient and Conjugate Gradient Methods
5.3 Preconditioned Conjugate Gradient Method
5.4 Krylov Subspace Methods for Nonsymmetric Equations
5.5 Multigrid Method
5.6 Nested Iterations
5.7 Space (Domain) Decomposition Methods
6 Beyond Coercivity, Consistency, and Conformity
6.1 General Variational Equations
6.2 Saddle Point Problems
6.3 Fluid Mechanics: Laminar Flows
7 Mixed and Nonconforming Discretization Methods
7.1 Nonconforming Finite Element Methods I: The Crouzeix–Raviart Element
7.2 Mixed Methods for the Darcy Equation
7.3 Mixed Methods for the Stokes Equation
7.4 Nonconforming Finite Element Methods II: Discontinuous Galerkin Methods
7.5 Hybridization
7.6 Local Mass Conservation and Flux Reconstruction
8 The Finite Volume Method
8.1 The Basic Idea of the Finite Volume Method
8.2 The Finite Volume Method for Linear Elliptic Differential Equations of Second Order on Triangular Grids
8.3 A Cell-oriented Finite Volume Method for Linear Elliptic Differential Equations of Second Order
8.4 Multipoint Flux Approximations
8.5 Finite Volume Methods in the Context of Mixed Finite Element Methods
8.6 Finite Volume Methods for the Stokes and Navier–Stokes Equations
9 Discretization Methods for Parabolic Initial Boundary Value Problems
9.1 Problem Setting and Solution Concept
9.2 Semidiscretization by the Vertical Method of Lines
9.3 Fully Discrete Schemes
9.4 Stability
9.5 High-Order One-Step and Multistep Methods
9.6 Exponential Integrators
9.7 The Maximum Principle
9.8 Order of Convergence Estimates in Space and Time
10 Discretization Methods for Convection-Dominated Problems
10.1 Standard Methods and Convection-Dominated Problems
10.2 The Streamline-Diffusion Method
10.3 Finite Volume Methods
10.4 The Lagrange–Galerkin Method
10.5 Algebraic Flux Correction and Limiting Methods
10.6 Slope Limitation Techniques
11 An Outlook to Nonlinear Partial Differential Equations
11.1 Nonlinear Problems and Iterative Methods
11.2 Fixed-Point Iterations
11.3 Newton's Method and Its Variants
11.4 Semilinear Boundary Value Problems for Elliptic and Parabolic Equations
11.5 Quasilinear Equations
11.6 Iterative Methods for Semilinear Differential Systems
11.7 Splitting Methods
A Appendices
A.1 Notation
A.2 Basic Concepts of Analysis
A.3 Basic Concepts of Linear Algebra
A.4 Some Definitions and Arguments of Linear Functional Analysis
A.5 Function Spaces
References: Textbooks and Monographs
References: Journal Papers and Other Resources
Index
