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Numerical methods for elliptic and parabolic partial differential equations Cover Image Book Book

Numerical methods for elliptic and parabolic partial differential equations

Knabner, Peter (author.).

Record details

  • ISBN: 9783030793845
  • ISBN: 3030793842
  • Physical Description: print
    xix, 802 pages ; 25 cm.
  • Edition: Second extended edition.
  • Publisher: Cham, Switzerland : Springer, [2021]

Content descriptions

Formatted Contents Note: Machine generated contents note: 0.1. 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 -- 1.1. Dirichlet Problem for the Poisson Equation -- 1.2. Finite Difference Method -- 1.3. Generalizations and Limitations of the Finite Difference Method -- 1.4. Maximum Principles and Stability -- 2.1. Variational Formulation for the Model Problem -- 2.2. Finite Element Method with Linear Elements -- 2.3. Stability and Convergence of the Finite Element Method -- 2.4. Implementation of the Finite Element Method: Part 1 -- 2.4.1. Preprocessor -- 2.4.2. Assembling -- 2.4.3. Realization of Dirichlet Boundary Conditions: Part 1 -- 2.4.4. Notes on Software -- 2.4.5. Testing Numerical Methods and Software -- 2.5. Solving Sparse Systems of Linear Equations by Direct Methods -- 3.1. Variational Equations and Sobolev Spaces -- 3.2. Elliptic Boundary Value Problems of Second Order -- 3.2.1. Variational Formulation of Special Cases -- 3.2.2. Example of a Boundary Value Problem of Fourth Order -- 3.2.3. Regularity of Boundary Value Problems -- 3.3. Element Types and Affine Equivalent Partitions -- 3.4. Convergence Rate Estimates -- 3.4.1. Energy Norm Estimates -- 3.4.2. Maximum Angle Condition on Triangles -- 3.4.3. L2 Error Estimates -- 3.5. Implementation of the Finite Element Method: Part 2 -- 3.5.1. Incorporation of Dirichlet Boundary Conditions: Part 2 -- 3.5.2. Numerical Quadrature -- 3.6. Convergence Rate Results in the Case of Quadrature and Interpolation -- 3.7. Condition Number of Finite Element Matrices -- 3.8. General Domains and Isoparametric Elements -- 3.9. Maximum Principle for Finite Element Methods -- 4.1. Grid Generation -- 4.1.1. Classification of Grids -- 4.1.2. Generation of Simplicial Grids -- 4.1.3. Generation of Quadrilateral and Hexahedral Grids -- 4.1.4. Grid Optimization -- 4.1.5. Grid Refinement -- 4.2. Posteriori Error Estimates -- 4.3. Convergence of Adaptive Methods -- 5.1. Linear Stationary Iterative Methods -- 5.1.1. General Theory -- 5.1.2. Classical Methods -- 5.1.3. Relaxation -- 5.1.4. SOR and Block-Iteration Methods -- 5.1.5. Extrapolation Methods -- 5.2. Gradient and Conjugate Gradient Methods -- 5.3. Preconditioned Conjugate Gradient Method -- 5.4. Krylov Subspace Methods for Nonsymmetric Systems of Equations -- 5.5. Multigrid Method -- 5.5.1. Idea of the Multigrid Method -- 5.5.2. Multigrid Method for Finite Element Discretizations -- 5.5.3. Effort and Convergence Behaviour -- 5.6. Nested Iterations -- 5.7. Space (Domain) Decomposition Methods -- 5.7.1. Preconditioning by Space Decomposition -- 5.7.2. Grid Decomposition Methods -- 5.7.3. Domain Decomposition Methods -- 6.1. General Variational Equations -- 6.2. Saddle Point Problems -- 6.2.1. Traces on Subsets of the Boundary -- 6.2.2. Mixed Variational Formulations -- 6.3. Fluid Mechanics: Laminar Flows -- 7.1. Nonconforming Finite Element Methods I: The Crouzeix-Raviart Element -- 7.2. Mixed Methods for the Darcy Equation -- 7.2.1. Dual Formulations in H(div; C) -- 7.2.2. Simplicial Finite Elements in H(div; -- 7.2.3. Finite Elements in H(div; n) on Quadrangles and Hexahedra -- 7.3. Mixed Methods for the Stokes Equation -- 7.4. Nonconforming Finite Element Methods II: Discontinuous Galerkin Methods -- 7.4.1. Interior Penalty Discontinuous Galerkin Methods -- 7.4.2. Additional Aspects of Interior Penalty and Related Methods -- 7.5. Hybridization -- 7.5.1. Hybridization in General -- 7.5.2. Convergence of the Multipliers for the Hybridized Mixed RT-Element Discretizations of the Darcy Equation -- 7.5.3. Hybrid Discontinuous Galerkin Methods -- 7.6. Local Mass Conservation and Flux Reconstruction -- 7.6.1. Approximation of Boundary Fluxes -- 7.6.2. Local Mass Conservation and Flux Reconstruction -- 8.1. Basic Idea of the Finite Volume Method -- 8.2. Finite Volume Method for Linear Elliptic Differential Equations of Second Order on Triangular Grids -- 8.2.1. Admissible Control Volumes -- 8.2.2. Finite Volume Discretization -- 8.2.3. Comparison with the Finite Element Method -- 8.2.4. Properties of the Discretization -- 8.3. Cell-oriented Finite Volume Method for Linear Elliptic Differential Equations of Second Order -- 8.3.1. One-Dimensional Case -- 8.3.2. Cell-centred Finite Volume Method on Polygonal/Polyhedral Grids -- 8.4. Multipoint Flux Approximations -- 8.5. Finite Volume Methods in the Context of Mixed Finite Element Methods -- 8.5.1. Problem and Its Mixed Formulation -- 8.5.2. Finite-Dimensional Aproximation -- 8.6. Finite Volume Methods for the Stokes and Navier-Stokes Equations -- 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 -- 9.5.1. One-Step Methods -- 9.5.2. Linear Multistep Methods -- 9.5.3. Discontinuous Galerkin Method (DGM) in Time -- 9.6. Exponential Integrators -- 9.7. Maximum Principle -- 9.8. Order of Convergence Estimates in Space and Time -- 10.1. Standard Methods and Convection-Dominated Problems -- 10.2. Streamline-Diffusion Method -- 10.3. Finite Volume Methods -- 10.4. Lagrange-Galerkin Method -- 10.5. Algebraic Flux Correction and Limiting Methods -- 10.5.1. Construction of a Low-order Semidiscrete Scheme -- 10.5.2. Fully Discrete System -- 10.5.3. Algebraic Flux Correction -- 10.5.4. Nonlinear AFC Scheme -- 10.5.5. Limiting Strategy -- 10.6. Slope Limitation Techniques -- 11.1. Nonlinear Problems and Iterative Methods -- 11.2. Fixed-Point Iterations -- 11.3. Newton's Method and Its Variants -- 11.3.1. Standard Form of Newton's Method -- 11.3.2. Modifications of Newton's Method -- 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 -- 11.7.1. Noniterative Operator Splitting -- 11.7.2. Iterative Operator Splitting -- 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.

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  • 2 of 2 copies available at University College of the North Libraries.

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Location Call Number / Copy Notes Barcode Shelving Location Holdable? Status Due Date
The Pas Campus Library QA 377 .K575 2021 (Text) 58500001125319 Stacks Volume hold Available -
The Pas Campus Library QA 377 .K57513 2021 (Text) 58500000073205 Stacks Volume hold Available -
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