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E-raamat: Elliptic Differential Equations: Theory and Numerical Treatment

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Elliptic Differential Equations: Theory and Numerical TreatmentSuurem pilt
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This book simultaneously presents the theory and the numerical treatment of elliptic boundary value problems, since an understanding of the theory is necessary for the numerical analysis of the discretisation. It first discusses the Laplace equation and its finite difference discretisation before addressing the general linear differential equation of second order. The variational formulation together with the necessary background from functional analysis provides the basis for the Galerkin and finite-element methods, which are explored in detail. A more advanced chapter leads the reader to the theory of regularity. Individual chapters are devoted to singularly perturbed as well as to elliptic eigenvalue problems. The book also presents the Stokes problem and its discretisation as an example of a saddle-point problem taking into account its relevance to applications in fluid dynamics.



This is the soft cover reprint of the very popular hardcover edition. The book offers a simultaneous presentation of the theory and of the numerical treatment of elliptic problems.
1 Partial Differential Equations and Their Classification Into Types
1(12)
1.1 Examples
1(4)
1.2 Classification of Second-Order Equations into Types
5(2)
1.3 Type Classification for Systems of First Order
7(2)
1.4 Characteristic Properties of the Different Types
9(3)
1.5 Literature
12(1)
2 The Potential Equation
13(16)
2.1 Posing the Problem
13(3)
2.2 Singularity Function
16(3)
2.3 Mean-Value Property and Maximum Principle
19(6)
2.4 Continuous Dependence on the Boundary Data
25(4)
3 The Poisson Equation
29(14)
3.1 Posing the Problem
29(1)
3.2 Representation of the Solution by the Green Function
30(3)
3.3 Existence of a Solution
33(5)
3.4 The Green Function for the Ball
38(1)
3.5 The Neumann Boundary-Value Problem
39(2)
3.6 The Integral Equation Method
41(2)
4 Difference Methods for the Poisson Equation
43(50)
4.1 Introduction: The One-Dimensional Case
44(2)
4.2 The Five-Point Formula
46(4)
4.3 M-matrices, Matrix Norms, Positive-Definite Matrices
50(9)
4.4 Properties of the Matrix Lh
59(7)
4.5 Convergence
66(3)
4.6 Discretisations of Higher Order
69(3)
4.7 The Discretisation of the Neumann Boundary-Value Problem
72(14)
4.7.1 One-Sided Difference for ∂u/∂n
73(4)
4.7.2 Symmetric Difference for ∂u/∂n
77(2)
4.7.3 Symmetric Difference for ∂u/∂n on an Offset Grid
79(1)
4.7.4 Proof of the Stability Theorem 4.62
79(7)
4.8 Discretisation in an Arbitrary Domain
86(7)
4.8.1 Shortley-Weller Approximation
86(4)
4.8.2 Interpolation in Near-Boundary Points
90(3)
5 General Boundary-Value Problems
93(26)
5.1 Dirichlet Boundary-Value Problems for Linear Differential Equations
93(13)
5.1.1 Posing the Problem
93(2)
5.1.2 Maximum Principle
95(3)
5.1.3 Uniqueness of the Solution and Continuous Dependence
98(2)
5.1.4 Difference Methods for the General Differential Equation of Second Order
100(5)
5.1.5 Green's Function
105(1)
5.2 General Boundary Conditions
106(7)
5.2.1 Formulating the Boundary-Value Problem
106(3)
5.2.2 Difference Methods for General Boundary Conditions
109(4)
5.3 Boundary Problems of Higher Order
113(6)
5.3.1 The Biharmonic Differential Equation
113(1)
5.3.2 General Linear Differential Equations of Order 2m
113(2)
5.3.3 Discretisation of the Biharmonic Differential Equation
115(4)
6 Tools from Functional Analysis
119(40)
6.1 Banach Spaces and Hilbert Spaces
119(6)
6.1.1 Normed Spaces
119(1)
6.1.2 Operators
120(1)
6.1.3 Banach Spaces
121(2)
6.1.4 Hilbert Spaces
123(2)
6.2 Sobolev Spaces
125(17)
6.2.1 L2(Ω)
125(2)
6.2.2 Hk(Ω) and Hk0(Ω)
127(3)
6.2.3 Fourier Transformation and Hk(Rn)
130(3)
6.2.4 Hs(Ω) for Real s ≥ 0
133(1)
6.2.5 Trace and Extension Theorems
134(8)
6.3 Dual Spaces
142(5)
6.3.1 Dual Space of a Normed Space
142(1)
6.3.2 Adjoint Operators
143(2)
6.3.3 Scales of Hilbert Spaces
145(2)
6.4 Compact Operators
147(4)
6.5 Bilinear Forms
151(8)
7 Variational Formulation
159(22)
7.1 Historical Remarks About the Dirichlet Principle
159(2)
7.2 Equations with Homogeneous Dirichlet Boundary Conditions
161(7)
7.2.1 Dirichlet Boundary Condition
161(1)
7.2.2 Weak Formulation
162(2)
7.2.3 Hmo(Ω)-Ellipticity
164(3)
7.2.4 Hmo(Ω)-Coercivity
167(1)
7.3 Inhomogeneous Dirichlet Boundary Conditions
168(2)
7.4 Natural Boundary Conditions
170(9)
7.4.1 Variation in Hm{Ω)
170(1)
7.4.2 Conormal Boundary Condition
171(2)
7.4.3 Oblique Boundary Condition
173(3)
7.4.4 Boundary Conditions for m ≥ 2
176(2)
7.4.5 Further Boundary Conditions
178(1)
7.5 Pseudo-Differential Equations
179(2)
8 The Finite-Element Method
181(82)
8.1 Historical Remarks
181(2)
8.2 The Ritz-Galerkin Method
183(11)
8.2.1 Basics
183(3)
8.2.2 Analysis of the Discrete Equation
186(4)
8.2.3 Solvability of the Discrete Problem
190(2)
8.2.4 Examples
192(2)
8.3 Error Estimates
194(6)
8.3.1 Quasi-Optimality
194(1)
8.3.2 Convergence of the Ritz-Galerkin Solutions
195(2)
8.3.3 Ritz Projection
197(1)
8.3.4 Further Stability and Error Estimates
198(2)
8.4 Finite Elements
200(13)
8.4.1 Introduction: Linear Elements for Ω = (a, b)
200(3)
8.4.2 Linear Elements for Ω ⊂ R2
203(3)
8.4.3 Bilinear Elements for Ω ⊂ R2
206(2)
8.4.4 Quadratic Elements for Ω ⊂ R2
208(1)
8.4.5 Elements for Ω ⊂ R3
209(1)
8.4.6 Handling of Side Conditions
210(3)
8.5 Error Estimates for Finite-Element Methods
213(11)
8.5.1 Preparations
213(3)
8.5.2 Properties of Sequences of Finite-Element Spaces
216(2)
8.5.3 H1-Estimates for Linear Elements
218(2)
8.5.4 L2 Estimates for Linear Elements
220(4)
8.6 Generalisations
224(6)
8.6.1 Error Estimates for Other Elements
224(1)
8.6.2 Finite Elements for Equations of Higher Order
225(2)
8.6.3 Finite Elements for Non-Polygonal Regions
227(3)
8.7 A-posteriori Error Estimates, Adaptivity
230(11)
8.7.1 A-posteriori Error Estimates
230(6)
8.7.2 Efficiency of the Finite-Element Method
236(1)
8.7.3 Adaptive Finite-Element Method
237(4)
8.8 Properties of the System Matrix
241(7)
8.8.1 Connection of L and Lh
241(1)
8.8.2 Equivalent Norms and Mass Matrix
241(3)
8.8.3 Inverse Estimate and Condition of L
244(2)
8.8.4 Element Matrices
246(1)
8.8.5 Positivity, Maximum Principle
247(1)
8.9 Further Remarks
248(15)
8.9.1 Mixed and Hybrid Finite-Element Methods
248(1)
8.9.2 Nonconforming Elements
249(2)
8.9.3 Inadmissible Triangulations
251(1)
8.9.4 Trefftz' Method
252(1)
8.9.5 Finite-Element Methods for Singular Solutions
253(1)
8.9.6 Hierarchical Bases
253(1)
8.9.7 Superconvergence
254(1)
8.9.8 Mortar Finite Elements
255(2)
8.9.9 Composite Finite Elements
257(1)
8.9.10 Related Discretisations
258(2)
8.9.11 Sparse Grids
260(3)
9 Regularity
263(48)
9.1 Solutions of the Boundary-Value Problem in Hs (Ω), s >m
263(22)
9.1.1 The Regularity Problem
264(2)
9.1.2 Regularity Theorems for Ω = Rn
266(8)
9.1.3 Regularity Theorems for Ω = Rn+
274(4)
9.1.4 Regularity Theorems for General Domains Ω ⊂ Rn
278(4)
9.1.5 Regularity for Convex Domains and Domains with Corners
282(3)
9.2 Regularity in the Interior
285(5)
9.2.1 Estimates
286(1)
9.2.2 Behaviour of the Singularity and Green's Function
286(4)
9.3 Regularity Properties of Difference Equations
290(21)
9.3.1 Discrete H1-Regularity
290(6)
9.3.2 Consistency
296(7)
9.3.3 Optimal Error Estimates
303(2)
9.3.4 Hmo, h+θ-Regularity for -1/2 < θ < 1/2
305(1)
9.3.5 H2h-Regularity
306(3)
9.3.6 Interior Regularity
309(2)
10 Special Differential Equations
311(18)
10.1 Differential Equations with Discontinuous Coefficients
311(5)
10.1.1 Formulation
311(3)
10.1.2 Finite-Element Discretisation
314(1)
10.1.3 Discretisation by Difference Schemes
315(1)
10.1.4 Discontinuous Coefficients of the First and Zeroth Derivatives
315(1)
10.2 A Singular Perturbation Problem
316(13)
10.2.1 The Convection-Diffusion Equation
316(2)
10.2.2 Stable Difference Schemes
318(3)
10.2.3 Finite Elements
321(8)
11 Elliptic Eigenvalue Problems
329(26)
11.1 Formulation of Eigenvalue Problems
329(2)
11.2 Finite-Element Discretisation
331(15)
11.2.1 Discretisation
331(2)
11.2.2 Qualitative Convergence Results
333(5)
11.2.3 Quantitative Convergence Results
338(5)
11.2.4 Consistent Problems
343(3)
11.3 Discretisation by Difference Methods
346(8)
11.4 Further Remarks
354(1)
12 Stokes Equations
355(26)
12.1 Elliptic Systems of Differential Equations
355(4)
12.2 Variational Formulation
359(12)
12.2.1 Weak Formulation of the Stokes Equations
359(1)
12.2.2 Saddle-Point Problems
360(3)
12.2.3 Existence and Uniqueness of the Solution of a Saddle-Point Problem
363(3)
12.2.4 Solvability and Regularity of the Stokes Problem
366(4)
12.2.5 A Vo-elliptic Variational Formulation of the Stokes Problem
370(1)
12.3 Finite-Element Method for the Stokes Problem
371(10)
12.3.1 Finite-Element Discretisation of a Saddle-Point Problem
371(2)
12.3.2 Stability Conditions
373(1)
12.3.3 Stable Finite-Element Spaces for the Stokes Problem
374(6)
12.3.4 Divergence-Free Elements
380(1)
A Solution of the Exercises
381(48)
Exercises of
Chapter 1
381(5)
Exercises of
Chapter 2
386(5)
Exercises of
Chapter 3
391(4)
Exercises of
Chapter 4
395(6)
Exercises of
Chapter 5
401(3)
Exercises of
Chapter 6
404(4)
Exercises of
Chapter 7
408(2)
Exercises of
Chapter 8
410(3)
Exercises of
Chapter 9
413(5)
Exercises of
Chapter 10
418(1)
Exercises of
Chapter 11
419(6)
Exercises of
Chapter 12
425(4)
References 429(14)
List of Symbols and Abbreviations 443(6)
Index 449
The author is a very well-known author of Springer, working in the field of numerical mathematics for partial differential equations and integral equations. He has published numerous books in the SSCM series, e.g., about the multi-grid method, about the numerical analysis of elliptic pdes, about iterative solution of large systems of equation, and a book in German about the technique of hierarchical matrices. Hackbusch is member of the editorial board of Springer' s book series "Advances in Numerical Mathematics", "The International Cryogenics Monograph Series" and "Springer Series of Computational Mathematics".
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