Finite elements and fast iterative solvers : with applications in incompressible fluid dynamics /

Finite elements and fast iterative solvers : with applications in incompressible fluid dynamics /

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Bibliographic Details
Author / Creator:Elman, Howard C.
Imprint:New York : Oxford University Press, 2005.
Description:xiii, 400 p. : ill. ; 24 cm.
Language:English
Series:Numerical mathematics and scientific computation
Subject:
Computational fluid dynamics
Differential equations, Partial.
Finite element method.
Differential equations, Partial.
Finite element method.
Computational fluid dynamics.
Format: Print Book
URL for this record:http://pi.lib.uchicago.edu/1001/cat/bib/5700659
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Other authors / contributors:Silvester, David J.
Wathen, Andrew J.
ISBN:9780198528678 (alk. paper)
019852868X (alk. paper)
9780198528685 (alk. paper)
0198528671 (alk. paper)
Notes:Includes bibliographical references (p. 382-395) and index.
Table of Contents:
  • 0. Models of incompressible fluid flow
  • 1. The Poisson equation
  • 1.1. Reference problems
  • 1.2. Weak formulation
  • 1.3. The Galerkin finite element method
  • 1.3.1. Triangular finite elements (R[superscript 2])
  • 1.3.2. Quadrilateral elements (R[superscript 2])
  • 1.3.3. Tetrahedral elements (R[superscript 3])
  • 1.3.4. Brick elements (R[superscript 3])
  • 1.4. Implementation aspects
  • 1.4.1. Triangular element matrices
  • 1.4.2. Quadrilateral element matrices
  • 1.4.3. Assembly of the Galerkin system
  • 1.5. Theory of errors
  • 1.5.1. A priori error bounds
  • 1.5.2. A posteriori error bounds
  • 1.6. Matrix properties
  • Problems
  • Computational exercises
  • 2. Solution of discrete Poisson problems
  • 2.1. The conjugate gradient method
  • 2.1.1. Convergence analysis
  • 2.1.2. Stopping criteria
  • 2.2. Preconditioning
  • 2.3. Singular systems are not a problem
  • 2.4. The Lanczos and minimum residual methods
  • 2.5. Multigrid
  • 2.5.1. Two-grid convergence theory
  • 2.5.2. Extending two-grid to multigrid
  • Problems
  • Computational exercises
  • 3. The convection-diffusion equation
  • 3.1. Reference problems
  • 3.2. Weak formulation and the convection term
  • 3.3. Approximation by finite elements
  • 3.3.1. The Galerkin finite element method
  • 3.3.2. The streamline diffusion method
  • 3.4. Theory of errors
  • 3.4.1. A priori error bounds
  • 3.4.2. A posteriori error bounds
  • 3.5. Matrix properties
  • 3.5.1. Computational molecules and Fourier analysis
  • 3.5.2. Analysis of difference equations
  • Discussion and bibliographical notes
  • Problems
  • Computational exercises
  • 4. Solution of discrete convection-diffusion problems
  • 4.1. Krylov subspace methods
  • 4.1.1. GMRES
  • 4.1.2. Biorthogonalization methods
  • 4.2. Preconditioning methods and splitting operators
  • 4.2.1. Splitting operators for convection-diffusion systems
  • 4.2.2. Matrix analysis of convergence
  • 4.2.3. Asymptotic analysis of convergence
  • 4.2.4. Practical considerations
  • 4.3. Multigrid
  • 4.3.1. Practical issues
  • 4.3.2. Tools of analysis: smoothing and approximation properties
  • 4.3.3. Smoothing
  • 4.3.4. Analysis
  • Discussion and bibliographical notes
  • Problems
  • Computational exercises
  • 5. The Stokes equations
  • 5.1. Reference problems
  • 5.2. Weak formulation
  • 5.3. Approximation using mixed finite elements
  • 5.3.1. Stable rectangular elements (Q[subscript 2]-Q[subscript 1], Q[subscript 2]-P[subscript -1], Q[subscript 2]-P[subscript 0])
  • 5.3.2. Stabilized rectangular elements (Q[subscript 1]-P[subscript 0], Q[subscript 1]-Q[subscript 1])
  • 5.3.3. Triangular elements
  • 5.3.4. Brick and tetrahedral elements
  • 5.4. Theory of errors
  • 5.4.1. A priori error bounds
  • 5.4.2. A posteriori error bounds
  • 5.5. Matrix properties
  • 5.5.1. Stable mixed approximation
  • 5.5.2. Stabilized mixed approximation
  • Discussion and bibliographical notes
  • Problems
  • Computational exercises
  • 6. Solution of discrete Stokes problems
  • 6.1. The preconditioned MINRES method
  • 6.2. Preconditioning
  • 6.2.1. General strategies for preconditioning
  • 6.2.2. Eigenvalue bounds
  • 6.2.3. Equivalent norms for MINRES
  • 6.2.4. MINRES convergence analysis
  • Discussion and bibliographical notes
  • Problems
  • Computational exercises
  • 7. The Navier-Stokes equations
  • 7.1. Reference problems
  • 7.2. Weak formulation and linearization
  • 7.2.1. Stability theory and bifurcation analysis
  • 7.2.2. Nonlinear iteration
  • 7.3. Mixed finite element approximation
  • 7.4. Theory of errors
  • 7.4.1. A priori error bounds
  • 7.4.2. A posteriori error bounds
  • Discussion and bibliographical notes
  • Problems
  • Computational exercises
  • 8. Solution of discrete Navier-Stokes problems
  • 8.1. General strategies for preconditioning
  • 8.2. Approximations to the Schur complement operator
  • 8.2.1. The pressure convection-diffusion preconditioner
  • 8.2.2. The least-squares commutator preconditioner
  • 8.3. Performance and analysis
  • 8.3.1. Ideal versions of the preconditioners
  • 8.3.2. Use of iterative methods for subproblems
  • 8.3.3. Convergence analysis
  • 8.3.4. Enclosed flow: singular systems are not a problem
  • 8.3.5. Relation to SIMPLE iteration
  • 8.4. Nonlinear iteration
  • Discussion and bibliographical notes
  • Problems
  • Computational exercises
  • Bibliography

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