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E-raamat: Numerical Methods for Elliptic and Parabolic Partial Differential Equations: With contributions by Andreas Rupp

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  • Formaat: PDF+DRM
  • Sari: Texts in Applied Mathematics 44
  • Ilmumisaeg: 19-Nov-2021
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030793852
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Numerical Methods for Elliptic and Parabolic Partial Differential Equations: With contributions by Andreas RuppSuurem pilt
  • Formaat: PDF+DRM
  • Sari: Texts in Applied Mathematics 44
  • Ilmumisaeg: 19-Nov-2021
  • Kirjastus: Springer Nature Switzerland AG
  • Keel: eng
  • ISBN-13: 9783030793852
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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.

Arvustused

This book has a large amount of new exercise problems that are uniformly distributed across the text. this book is a very nice text which will serve well for the undergraduate as well as graduate students and will also become a ready reference for scholars. (Murli M. Gupta, Mathematical Reviews, April, 2023)



Many of the SIAM Review readership will be interested in NMEPPDE from the standpoint of self-study or classroom education. NMEPPDE offers the applied mathematics reader nearly a single point of entry to our broad and challenging area. a bit of open space on the bookshelf could profitably be well filled with a copy of NMEPPDE. (Robert C. Kirby, SIAM Review, Vol. 65 (1), March, 2023)

Preface to the Second English Edition v
From the Preface to the First English Edition xi
Preface to the German Edition xiii
0 For Example: Modelling Processes in Porous Media with Differential Equations 1(18)
0.1 The Basic Partial Differential Equation Models
1(4)
0.2 Reactions and Transport in Porous Media
5(2)
0.3 Fluid Flow in Porous Media
7(3)
0.4 Reactive Solute Transport in Porous Media
10(3)
0.5 Boundary and Initial Value Problems
13(6)
1 For the Beginning: The Finite Difference Method for the Poisson Equation 19(32)
1.1 The Dirichlet Problem for the Poisson Equation
19(2)
1.2 The Finite Difference Method
21(9)
1.3 Generalizations and Limitations of the Finite Difference Method
30(11)
1.4 Maximum Principles and Stability
41(10)
2 The Finite Element Method for the Poisson Equation 51(60)
2.1 Variational Formulation for the Model Problem
51(10)
2.2 The Finite Element Method with Linear Elements
61(14)
2.3 Stability and Convergence of the Finite Element Method
75(7)
2.4 The Implementation of the Finite Element Method: Part 1
82(10)
2.4.1 Preprocessor
82(2)
2.4.2 Assembling
84(5)
2.4.3 Realization of Dirichlet Boundary Conditions: Part 1
89(1)
2.4.4 Notes on Software
90(1)
2.4.5 Testing Numerical Methods and Software
91(1)
2.5 Solving Sparse Systems of Linear Equations by Direct Methods
92(19)
3 The Finite Element Method for Linear Elliptic Boundary Value Problems of Second Order 111(94)
3.1 Variational Equations and Sobolev Spaces
111(7)
3.2 Elliptic Boundary Value Problems of Second Order
118(16)
3.2.1 Variational Formulation of Special Cases
120(11)
3.2.2 An Example of a Boundary Value Problem of Fourth Order
131(1)
3.2.3 Regularity of Boundary Value Problems
132(2)
3.3 Element Types and Affine Equivalent Partitions
134(19)
3.4 Convergence Rate Estimates
153(18)
3.4.1 Energy Norm Estimates
153(11)
3.4.2 The Maximum Angle Condition on Triangles
164(4)
3.4.3 L2 Error Estimates
168(3)
3.5 The Implementation of the Finite Element Method: Part 2
171(9)
3.5.1 Incorporation of Dirichlet Boundary Conditions: Part 2
171(3)
3.5.2 Numerical Quadrature
174(6)
3.6 Convergence Rate Results in the Case of Quadrature and Interpolation
180(8)
3.7 The Condition Number of Finite Element Matrices
188(5)
3.8 General Domains and Isoparametric Elements
193(6)
3.9 The Maximum Principle for Finite Element Methods
199(6)
4 Grid Generation and A Posteriori Error Estimation 205(30)
4.1 Grid Generation
205(11)
4.1.1 Classification of Grids
205(1)
4.1.2 Generation of Simplicial Grids
206(3)
4.1.3 Generation of Quadrilateral and Hexahedral Grids
209(1)
4.1.4 Grid Optimization
210(1)
4.1.5 Grid Refinement
211(5)
4.2 A Posteriori Error Estimates
216(12)
4.3 Convergence of Adaptive Methods
228(7)
5 Iterative Methods for Systems of Linear Equations 235(106)
5.1 Linear Stationary Iterative Methods
237(18)
5.1.1 General Theory
237(2)
5.1.2 Classical Methods
239(5)
5.1.3 Relaxation
244(3)
5.1.4 SOR and Block-Iteration Methods
247(5)
5.1.5 Extrapolation Methods
252(3)
5.2 Gradient and Conjugate Gradient Methods
255(11)
5.3 Preconditioned Conjugate Gradient Method
266(6)
5.4 Krylov Subspace Methods for Nonsymmetric Systems of Equations
272(14)
5.5 The Multigrid Method
286(21)
5.5.1 The Idea of the Multigrid Method
286(2)
5.5.2 Multigrid Method for Finite Element Discretizations
288(8)
5.5.3 Effort and Convergence Behaviour
296(11)
5.6 Nested Iterations
307(3)
5.7 Space (Domain) Decomposition Methods
310(31)
5.7.1 Preconditioning by Space Decomposition
311(7)
5.7.2 Grid Decomposition Methods
318(9)
5.7.3 Domain Decomposition Methods
327(14)
6 Beyond Coercivity, Consistency, and Conformity 341(52)
6.1 General Variational Equations
341(15)
6.2 Saddle Point Problems
356(26)
6.2.1 Traces on Subsets of the Boundary
358(3)
6.2.2 Mixed Variational Formulations
361(21)
6.3 Fluid Mechanics: Laminar Flows
382(11)
7 Mixed and Nonconforming Discretization Methods 393(94)
7.1 Nonconforming Finite Element Methods I: The Crouzeix-Raviart Element
393(14)
7.2 Mixed Methods for the Darcy Equation
407(18)
7.2.1 Dual Formulations in H(div; C)
407(1)
7.2.2 Simplicial Finite Elements in H(div;
408(13)
7.2.3 Finite Elements in H(div; n) on Quadrangles and Hexahedra
421(4)
7.3 Mixed Methods for the Stokes Equation
425(15)
7.4 Nonconforming Finite Element Methods II: Discontinuous Galerkin Methods
440(20)
7.4.1 Interior Penalty Discontinuous Galerkin Methods
441(12)
7.4.2 Additional Aspects of Interior Penalty and Related Methods
453(7)
7.5 Hybridization
460(15)
7.5.1 Hybridization in General
460(6)
7.5.2 Convergence of the Multipliers for the Hybridized Mixed RT-Element Discretizations of the Darcy Equation
466(5)
7.5.3 Hybrid Discontinuous Galerkin Methods
471(4)
7.6 Local Mass Conservation and Flux Reconstruction
475(12)
7.6.1 Approximation of Boundary Fluxes
475(4)
7.6.2 Local Mass Conservation and Flux Reconstruction
479(8)
8 The Finite Volume Method 487(70)
8.1 The Basic Idea of the Finite Volume Method
489(5)
8.2 The Finite Volume Method for Linear Elliptic Differential Equations of Second Order on Triangular Grids
494(23)
8.2.1 Admissible Control Volumes
494(3)
8.2.2 Finite Volume Discretization
497(7)
8.2.3 Comparison with the Finite Element Method
504(4)
8.2.4 Properties of the Discretization
508(9)
8.3 A Cell-oriented Finite Volume Method for Linear Elliptic Differential Equations of Second Order
517(15)
8.3.1 The One-Dimensional Case
517(12)
8.3.2 A Cell-centred Finite Volume Method on Polygonal/Polyhedral Grids
529(3)
8.4 Multipoint Flux Approximations
532(4)
8.5 Finite Volume Methods in the Context of Mixed Finite Element Methods
536(8)
8.5.1 The Problem and Its Mixed Formulation
536(2)
8.5.2 The Finite-Dimensional Aproximation
538(6)
8.6 Finite Volume Methods for the Stokes and Navier-Stokes Equations
544(13)
9 Discretization Methods for Parabolic Initial Boundary Value Problems 557(104)
9.1 Problem Setting and Solution Concept
557(14)
9.2 Semidiscretization by the Vertical Method of Lines
571(30)
9.3 Fully Discrete Schemes
601(6)
9.4 Stability
607(12)
9.5 High-Order One-Step and Multistep Methods
619(10)
9.5.1 One-Step Methods
619(4)
9.5.2 Linear Multistep Methods
623(3)
9.5.3 Discontinuous Galerkin Method (DGM) in Time
626(3)
9.6 Exponential Integrators
629(9)
9.7 The Maximum Principle
638(10)
9.8 Order of Convergence Estimates in Space and Time
648(13)
10 Discretization Methods for Convection-Dominated Problems 661(36)
10.1 Standard Methods and Convection-Dominated Problems
661(8)
10.2 The Streamline-Diffusion Method
669(8)
10.3 Finite Volume Methods
677(4)
10.4 The Lagrange-Galerkin Method
681(2)
10.5 Algebraic Flux Correction and Limiting Methods
683(10)
10.5.1 Construction of a Low-order Semidiscrete Scheme
685(3)
10.5.2 The Fully Discrete System
688(1)
10.5.3 Algebraic Flux Correction
689(2)
10.5.4 The Nonlinear AFC Scheme
691(1)
10.5.5 A Limiting Strategy
692(1)
10.6 Slope Limitation Techniques
693(4)
11 An Outlook to Nonlinear Partial Differential Equations 697(56)
11.1 Nonlinear Problems and Iterative Methods
697(6)
11.2 Fixed-Point Iterations
703(4)
11.3 Newton's Method and Its Variants
707(12)
11.3.1 The Standard Form of Newton's Method
707(5)
11.3.2 Modifications of Newton's Method
712(7)
11.4 Semilinear Boundary Value Problems for Elliptic and Parabolic Equations
719(14)
11.5 Quasilinear Equations
733(4)
11.6 Iterative Methods for Semilinear Differential Systems
737(3)
11.7 Splitting Methods
740(13)
11.7.1 Noniterative Operator Splitting
740(7)
11.7.2 Iterative Operator Splitting
747(6)
A Appendices 753(24)
A.1 Notation
753(6)
A.2 Basic Concepts of Analysis
759(1)
A.3 Basic Concepts of Linear Algebra
760(6)
A.4 Some Definitions and Arguments of Linear Functional Analysis
766(6)
A.5 Function Spaces
772(5)
References: Textbooks and Monographs 777(4)
References: Journal Papers and Other Resources 781(10)
Index 791
Peter Knabner is Professor emeritus at the University of Erlangen-Nürnberg, where he has led the chair Applied Mathematics I from 1994 to 2020, and also guest professor at the cluster of excellence SimTech of the University of Stuttgart. Knabnerresearch  is focussed on the derivation, analysis and numerical approximation of mathematical models for flow and transport in porous media. with applications in science and technology, in particular in hydrogeology.  After the study of Mathematics and Computer Science at the Freie Universität Berlin (diploma in 1972) he earned a PhD from the University of Augsburg in 1983, where he also received a higher doctoral degree (habilitation) in 1988.  Peter Knabner is author of more than 180 peer-reviewed publications in applied analysis, numerical mathematics and geohydrology. He is author and co-author of 13 research monographs and textbooks in German and English. 





Lutz Angermann is Professor of Numerical Mathematics at the Department of Mathematics of the Clausthal University of Technology since 2001. His research is concerned with the development and mathematical analysis of numerical methods for solving partial differential equations with special interests in finite volume and finite element methods and their application to problems in Physics and Engineering. After the study of Mathematics at the State University of Kharkov (now V.N. Karazin Kharkiv National University, Ukraine) he earned a PhD from the University of Technology at Dresden in 1987. The University of Erlangen-Nürnberg awarded him a higher doctoral degree (habilitation) in 1995. From 1998 to 2001, he held the post of an Associate Professor of Numerical Mathematics at the University of Magdeburg. He has authored or co-authored about 100 scientific papers, among them four books as co-author, and he edited two books.
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