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E-raamat: Adaptive Numerical Solution of PDEs

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  • Formaat: 432 pages
  • Sari: De Gruyter Textbook
  • Ilmumisaeg: 31-Aug-2012
  • Kirjastus: De Gruyter
  • Keel: eng
  • ISBN-13: 9783110283112
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Adaptive Numerical Solution of PDEsSuurem pilt
  • Formaat: 432 pages
  • Sari: De Gruyter Textbook
  • Ilmumisaeg: 31-Aug-2012
  • Kirjastus: De Gruyter
  • Keel: eng
  • ISBN-13: 9783110283112
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"Numerical mathematics is a subtopic of scientific computing. The focus lies on the efficiency of algorithms, i.e. speed, reliability, and robustness. This leads to adaptive algorithms. The theoretical derivation und analyses of algorithms are kept as elementary as possible in this book; the needed sligtly advanced mathematical theory is summarized in the appendix. Numerous figures and illustrating examples explain the complex data, as non-trivial examples serve problems from nanotechnology, chirurgy, and physiology. The book addresses students as well as practitioners in mathematics, natural sciences, and engineering. It is designed as a textbook but also suitable for self study. "

Peter Deuflhard and Martin Weiser, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany.
Preface v
Outline 1(4)
1 Elementary Partial Differential Equations
5(29)
1.1 Laplace and Poisson Equation
5(10)
1.1.1 Boundary Value Problems
6(4)
1.1.2 Initial Value Problem
10(2)
1.1.3 Eigenvalue Problem
12(3)
1.2 Diffusion Equation
15(3)
1.3 Wave Equation
18(5)
1.4 Schrodinger Equation
23(3)
1.5 Helmholtz Equation
26(3)
1.5.1 Boundary Value Problems
26(1)
1.5.2 Time-harmonic Differential Equations
27(2)
1.6 Classification
29(2)
1.7 Exercises
31(3)
2 Partial Differential Equations in Science and Technology
34(28)
2.1 Electrodynamics
34(6)
2.1.1 Maxwell Equations
34(3)
2.1.2 Optical Model Hierarchy
37(3)
2.2 Fluid Dynamics
40(12)
2.2.1 Euler Equations
41(3)
2.2.2 Navier-Stokes Equations
44(5)
2.2.3 Prandtl's Boundary Layer
49(2)
2.2.4 Porous Media Equation
51(1)
2.3 Elastomechanics
52(7)
2.3.1 Basic Concepts of Nonlinear Elastomechanics
52(4)
2.3.2 Linear Elastomechanics
56(3)
2.4 Exercises
59(3)
3 Finite Difference Methods for Poisson Problems
62(24)
3.1 Discretization of Standard Problem
62(9)
3.1.1 Discrete Boundary Value Problems
63(5)
3.1.2 Discrete Eigenvalue Problem
68(3)
3.2 Approximation Theory on Uniform Grids
71(7)
3.2.1 Discretization Error in L2
73(3)
3.2.2 Discretization Error in L∞
76(2)
3.3 Discretization on Nonuniform Grids
78(5)
3.3.1 One-dimensional Special Case
78(2)
3.3.2 Curved Boundaries
80(3)
3.4 Exercises
83(3)
4 Galerkin Methods
86(57)
4.1 General Scheme
86(9)
4.1.1 Weak Solutions
86(3)
4.1.2 Ritz Minimization for Boundary Value Problems
89(4)
4.1.3 Rayleigh-Ritz Minimization for Eigenvalue Problems
93(2)
4.2 Spectral Methods
95(13)
4.2.1 Realization by Orthogonal Systems
96(4)
4.2.2 Approximation Theory
100(3)
4.2.3 Adaptive Spectral Methods
103(5)
4.3 Finite Element Methods
108(20)
4.3.1 Meshes and Finite Element Spaces
108(3)
4.3.2 Elementary Finite Elements
111(10)
4.3.3 Realization of Finite Elements
121(7)
4.4 Approximation Theory for Finite Elements
128(11)
4.4.1 Boundary Value Problems
128(3)
4.4.2 Eigenvalue Problems
131(5)
4.4.3 Angle Condition for Nonuniform Meshes
136(3)
4.5 Exercises
139(4)
5 Numerical Solution of Linear Elliptic Grid Equations
143(60)
5.1 Direct Elimination Methods
144(6)
5.1.1 Symbolic Factorization
145(2)
5.1.2 Frontal Solvers
147(3)
5.2 Matrix Decomposition Methods
150(6)
5.2.1 Jacobi Method
152(2)
5.2.2 Gauss-Seidel Method
154(2)
5.3 Conjugate Gradient Method
156(14)
5.3.1 CG-Method as Galerkin Method
156(3)
5.3.2 Preconditioning
159(4)
5.3.3 Adaptive PCG-method
163(2)
5.3.4 A CG-variant for Eigenvalue Problems
165(5)
5.4 Smoothing Property of Iterative Solvers
170(10)
5.4.1 Illustration for the Poisson Model Problem
170(4)
5.4.2 Spectral Analysis for Jacobi Method
174(1)
5.4.3 Smoothing Theorems
175(5)
5.5 Iterative Hierarchical Solvers
180(14)
5.5.1 Classical Multigrid Methods
182(8)
5.5.2 Hierarchical-basis Method
190(3)
5.5.3 Comparison with Direct Hierarchical Solvers
193(1)
5.6 Power Optimization of a Darrieus Wind Generator
194(6)
5.7 Exercises
200(3)
6 Construction of Adaptive Hierarchical Meshes
203(43)
6.1 A Posteriori Error Estimators
203(20)
6.1.1 Residual Based Error Estimator
206(5)
6.1.2 Triangle Oriented Error Estimators
211(4)
6.1.3 Gradient Recovery
215(4)
6.1.4 Hierarchical Error Estimators
219(3)
6.1.5 Goal-oriented Error Estimation
222(1)
6.2 Adaptive Mesh Refinement
223(10)
6.2.1 Equilibration of Local Discretization Errors
224(5)
6.2.2 Refinement Strategies
229(4)
6.2.3 Choice of Solvers on Adaptive Hierarchical Meshes
233(1)
6.3 Convergence on Adaptive Meshes
233(7)
6.3.1 A Convergence Proof
234(2)
6.3.2 An Example with a Reentrant Corner
236(4)
6.4 Design of a Plasmon-Polariton Waveguide
240(4)
6.5 Exercises
244(2)
7 Adaptive Multigrid Methods for Linear Elliptic Problems
246(64)
7.1 Subspace Correction Methods
246(25)
7.1.1 Basic Principle
247(3)
7.1.2 Sequential Subspace Correction Methods
250(5)
7.1.3 Parallel Subspace Correction Methods
255(4)
7.1.4 Overlapping Domain Decomposition Methods
259(7)
7.1.5 Higher-order Finite Elements
266(5)
7.2 Hierarchical Space Decompositions
271(11)
7.2.1 Decomposition into Hierarchical Bases
272(6)
7.2.2 L2-orthogonal Decomposition: BPX
278(4)
7.3 Multigrid Methods Revisited
282(7)
7.3.1 Additive Multigrid Methods
282(4)
7.3.2 Multiplicative Multigrid Methods
286(3)
7.4 Cascadic Multigrid Methods
289(11)
7.4.1 Theoretical Derivation
289(6)
7.4.2 Adaptive Realization
295(5)
7.5 Eigenvalue Problem Solvers
300(6)
7.5.1 Linear Multigrid Method
301(2)
7.5.2 Rayleigh Quotient Multigrid Method
303(3)
7.6 Exercises
306(4)
8 Adaptive Solution of Nonlinear Elliptic Problems
310(23)
8.1 Discrete Newton Methods for Nonlinear Grid Equations
311(8)
8.1.1 Exact Newton Methods
312(4)
8.1.2 Inexact Newton-PCG Methods
316(3)
8.2 Inexact Newton-Multigrid Methods
319(9)
8.2.1 Hierarchical Grid Equations
319(2)
8.2.2 Realization of Adaptive Algorithm
321(4)
8.2.3 An Elliptic Problem Without a Solution
325(3)
8.3 Operation Planning in Facial Surgery
328(3)
8.4 Exercises
331(2)
9 Adaptive Integration of Parabolic Problems
333(47)
9.1 Time Discretization of Stiff Differential Equations
333(20)
9.1.1 Linear Stability Theory
334(6)
9.1.2 Linearly Implicit One-step Methods
340(7)
9.1.3 Order Reduction
347(6)
9.2 Space-time Discretization of Parabolic PDEs
353(21)
9.2.1 Adaptive Method of Lines
354(8)
9.2.2 Adaptive Method of Time Layers
362(9)
9.2.3 Goal-oriented Error Estimation
371(3)
9.3 Electrical Excitation of the Heart Muscle
374(4)
9.3.1 Mathematical Models
374(1)
9.3.2 Numerical Simulation
375(3)
9.4 Exercises
378(2)
A Appendix
380(18)
A.1 Fourier Analysis and Fourier Transform
380(1)
A.2 Differential Operators in R3
381(2)
A.3 Integral Theorems
383(4)
A.4 Delta-Distribution and Green's Functions
387(5)
A.5 Sobolev Spaces
392(5)
A.6 Optimality Conditions
397(1)
B Software
398(3)
B.1 Adaptive Finite Element Codes
398(1)
B.2 Direct Solvers
399(1)
B.3 Nonlinear Solvers
399(2)
Bibliography 401(14)
Index 415
Peter Deuflhard and Martin Weiser, Konrad-Zuse-Zentrum für Informationstechnik Berlin, Germany.
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