Robust numerical methods for singularly perturbed differential equations : convection-diffusion-reaction and flow problems /

Robust numerical methods for singularly perturbed differential equations : convection-diffusion-reaction and flow problems /

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Bibliographic Details
Author / Creator:Roos, Hans-Görg, 1949-
Imprint:Berlin : Springer, 2008.
Description:1 online resource (xiv, 604 p.)
Language:English
Series:Springer series in computational mechanics, 0179-3632 ; 24
Springer series in computational mechanics ; 24.
Subject:
Numerical analysis.
Singular perturbations (Mathematics)
Boundary value problems.
Navier-Stokes equations.
Finite element method.
Boundary value problems.
Finite element method.
Navier-Stokes equations.
Numerical analysis.
Singular perturbations (Mathematics)
Format: E-Resource Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/8887019
Hidden Bibliographic Details
ISBN:9783540344674
3540344675
Notes:Includes bibliographical references (pp551-597) and index.
Description based on print version record.
Other form:Print version: Roos, Hans-Görg, 1949- Robust numerical methods for singularly perturbed differential equations. Berlin : Springer, 2008 9783540344667 3540344667
Table of Contents:
  • Notation
  • Introduction
  • Part I. Ordinary Differential Equations
  • 1. The Analytical Behaviour of Solutions
  • 1.1. Linear Second-Order Problems Without Turning Points
  • 1.1.1. Asymptotic Expansions
  • 1.1.2. The Green's Function and Stability Estimates
  • 1.1.3. A Priori Estimates for Derivatives and Solution Decomposition
  • 1.2. Linear Second-Order Turning-Point Problems
  • 1.3. Quasilinear Problems
  • 1.4. Linear Higher-Order Problems and Systems
  • 1.4.1. Asymptotic Expansions for Higher-Order Problems
  • 1.4.2. A Stability Result
  • 1.4.3. Systems of Ordinary Differential Equations
  • 2. Numerical Methods for Second-Order Boundary Value Problems
  • 2.1. Finite Difference Methods on Equidistant Meshes
  • 2.1.1. Classical Convergence Theory for Central Differencing
  • 2.1.2. Upwind Schemes
  • 2.1.3. The Concept of Uniform Convergence
  • 2.1.4. Uniformly Convergent Schemes of Higher Order
  • 2.1.5. Linear Turning-Point Problems
  • 2.1.6. Some Nonlinear Problems
  • 2.2. Finite Element Methods on Standard Meshes
  • 2.2.1. Basic Results for Standard Finite Element Methods
  • 2.2.2. Upwind Finite Elements
  • 2.2.3. Stabilized Higher-Order Methods
  • 2.2.4. Variational Multiscale and Differentiated Residual Methods
  • 2.2.5. Uniformly Convergent Finite Element Methods
  • 2.3. Finite Volume Methods
  • 2.4. Finite Difference Methods on Layer-adapted Grids
  • 2.4.1. Graded Meshes
  • 2.4.2. Piecewise Equidistant Meshes
  • 2.5. Adaptive Strategies Based on Finite Differences
  • Part II. Parabolic Initial-Boundary Value Problems in One Space Dimension
  • 1. Introduction
  • 2. Analytical Behaviour of Solutions
  • 2.1. Existence, Uniqueness, Comparison Principle
  • 2.2. Asymptotic Expansions and Bounds on Derivatives
  • 3. Finite Difference Methods
  • 3.1. First-Order Problems
  • 3.1.1. Consistency
  • 3.1.2. Stability
  • 3.1.3. Convergence in L[subscript 2]
  • 3.2. Convection-Diffusion Problems
  • 3.2.1. Consistency and Stability
  • 3.2.2. Convergence
  • 3.3. Polynomial Schemes
  • 3.4. Uniformly Convergent Methods
  • 3.4.1. Exponential Fitting in Space
  • 3.4.2. Layer-Adapted Tensor-Product Meshes
  • 3.4.3. Reaction-Diffusion Problems
  • 4. Finite Element Methods
  • 4.1. Space-Based Methods
  • 4.1.1. Polynomial Upwinding
  • 4.1.2. Uniformly Convergent Schemes
  • 4.1.3. Local Error Estimates
  • 4.2. Subcharacteristic-Based Methods
  • 4.2.1. SDFEM in Space-Time
  • 4.2.2. Explicit Galerkin Methods
  • 4.2.3. Eulerian-Lagrangian Methods
  • 5. Two Adaptive Methods
  • 5.1. Streamline Diffusion Methods
  • 5.2. Moving Mesh Methods (r-refinement)
  • Part III. Elliptic and Parabolic Problems in Several Space Dimensions
  • 1. Analytical Behaviour of Solutions
  • 1.1. Classical and Weak Solutions
  • 1.2. The Reduced Problem
  • 1.3. Asymptotic Expansions and Boundary Layers
  • 1.4. A Priori Estimates and Solution Decomposition
  • 2. Finite Difference Methods
  • 2.1. Finite Difference Methods on Standard Meshes
  • 2.1.1. Exponential Boundary Layers
  • 2.1.2. Parabolic Boundary Layers
  • 2.2. Layer-Adapted Meshes
  • 2.2.1. Exponential Boundary Layers
  • 2.2.2. Parabolic Layers
  • 3. Finite Element Methods
  • 3.1. Inverse-Monotonicity-Preserving Methods Based on Finite Volume Ideas
  • 3.2. Residual-Based Stabilizations
  • 3.2.1. Streamline Diffusion Finite Element Method (SDFEM)
  • 3.2.2. Galerkin Least Squares Finite Element Method (GLSFEM)
  • 3.2.3. Residual-Free Bubbles
  • 3.3. Adding Symmetric Stabilizing Terms
  • 3.3.1. Local Projection Stabilization
  • 3.3.2. Continuous Interior Penalty Stabilization
  • 3.4. The Discontinuous Galerkin Finite Element Method
  • 3.4.1. The Primal Formulation for a Reaction-Diffusion Problem
  • 3.4.2. A First-Order Hyperbolic Problem
  • 3.4.3. dGFEM Error Analysis for Convection-Diffusion Problems
  • 3.5. Uniformly Convergent Methods
  • 3.5.1. Operator-Fitted Methods
  • 3.5.2. Layer-Adapted Meshes
  • 3.6. Adaptive Methods
  • 3.6.1. Adaptive Finite Element Methods for Non-Singularly Perturbed Elliptic Problems: an Introduction
  • 3.6.2. Robust and Semi-Robust Residual Type Error Estimators
  • 3.6.3. A Variant of the DWR Method for Streamline Diffusion
  • 4. Time-Dependent Problems
  • 4.1. Analytical Behavior of Solutions
  • 4.2. Finite Difference Methods
  • 4.3. Finite Element Methods
  • Part IV. The Incompressible Navier-Stokes Equations
  • 1. Existence and Uniqueness Results
  • 2. Upwind Finite Element Method
  • 3. Higher-Order Methods of Streamline Diffusion Type
  • 3.1. The Oseen Problem
  • 3.2. The Navier-Stokes Problem
  • 4. Local Projection Stabilization for Equal-Order Interpolation
  • 4.1. Local Projection Stabilization in an Abstract Setting
  • 4.2. Convergence Analysis
  • 4.2.1. The Special Interpolant
  • 4.2.2. Stability
  • 4.2.3. Consistency Error
  • 4.2.4. A priori Error Estimate
  • 4.3. Local Projection onto Coarse-Mesh Spaces
  • 4.3.1. Simplices
  • 4.3.2. Quadrilaterals and Hexahedra
  • 4.4. Schemes Based on Enrichment of Approximation Spaces
  • 4.4.1. Simplices
  • 4.4.2. Quadrilaterals and Hexahedra
  • 4.5. Relationship to Subgrid Modelling
  • 4.5.1. Two-Level Approach with Piecewise Linear Elements
  • 4.5.2. Enriched Piecewise Linear Elements
  • 4.5.3. Spectral Equivalence of the Stabilizing Terms on Simplices
  • 5. Local Projection Method for Inf-Sup Stable Elements
  • 5.1. Discretization by Inf-Sup Stable Elements
  • 5.2. Stability and Consistency
  • 5.3. Convergence
  • 5.3.1. Methods of Order r in the Case [sigma] > 0
  • 5.3.2. Methods of Order r in the Case [sigma greater than or equal] 0
  • 5.3.3. Methods of Order r + 1/2
  • 6. Mass Conservation for Coupled Flow-Transport Problems
  • 6.1. A Model Problem
  • 6.2. Continuous and Discrete Mass Conservation
  • 6.3. Approximated Incompressible Flows
  • 6.4. Mass-Conservative Methods
  • 6.4.1. Higher-Order Flow Approximation
  • 6.4.2. Post-Processing of the Discrete Velocity
  • 6.4.3. Scott-Vogelius Elements
  • 7. Adaptive Error Control
  • References
  • Index

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