Numerical Methods for Elliptic and Parabolic Partial Differential Equations

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