Muutke küpsiste eelistusi

Numerical Methods for Nonlinear Elliptic Differential Equations: A Synopsis [Kõva köide]

(University of Marburg)
  • Formaat: Hardback, 776 pages, kõrgus x laius x paksus: 253x178x44 mm, kaal: 1675 g, 67 b/w line illustrations
  • Sari: Numerical Mathematics and Scientific Computation
  • Ilmumisaeg: 07-Oct-2010
  • Kirjastus: Oxford University Press
  • ISBN-10: 0199577048
  • ISBN-13: 9780199577040
Teised raamatud teemal:
Numerical Methods for Nonlinear Elliptic Differential Equations: A SynopsisSuurem pilt
  • Formaat: Hardback, 776 pages, kõrgus x laius x paksus: 253x178x44 mm, kaal: 1675 g, 67 b/w line illustrations
  • Sari: Numerical Mathematics and Scientific Computation
  • Ilmumisaeg: 07-Oct-2010
  • Kirjastus: Oxford University Press
  • ISBN-10: 0199577048
  • ISBN-13: 9780199577040
Teised raamatud teemal:
Püsilink: https://www.kriso.ee/db/9780199577040.html
Märksõnad:
Nonlinear elliptic problems play an increasingly important role in mathematics, science and engineering, creating an exciting interplay between the subjects. This is the first and only book to prove in a systematic and unifying way, stability, convergence and computing results for the different numerical methods for nonlinear elliptic problems. The proofs use linearization, compact perturbation of the coercive principal parts, or monotone operator techniques, and approximation theory. Examples are given for linear to fully nonlinear problems (highest derivatives occur nonlinearly) and for the most important space discretization methods: conforming and nonconforming finite element, discontinuous Galerkin, finite difference, wavelet (and, in a volume to follow, spectral and meshfree) methods. A number of specific long open problems are solved here: numerical methods for fully nonlinear elliptic problems, wavelet and meshfree methods for nonlinear problems, and more general nonlinear boundary conditions. We apply it to all these problems and methods, in particular to eigenvalues, monotone operators, quadrature approximations, and Newton methods. Adaptivity is discussed for finite element and wavelet methods.

The book has been written for graduate students and scientists who want to study and to numerically analyze nonlinear elliptic differential equations in Mathematics, Science and Engineering. It can be used as material for graduate courses or advanced seminars.

Arvustused

To read this book is really an enjoyment and goldmine for every expert as well as for every well-interested and well-educated student. * Zentralblatt MATH, 2011 *

Preface xiv
PART I ANALYTICAL RESULTS
1 From linear to nonlinear equations, fundamental results
3(29)
1.1 Introduction
3(1)
1.2 Linear versus nonlinear models
3(7)
1.3 Examples for nonlinear partial differential equations
10(3)
1.4 Fundamental results
13(19)
1.4.1 Linear operators and functionals in Banach spaces
13(5)
1.4.2 Inequalities and Lp(Ω) spaces
18(2)
1.4.3 Holder and Sobolev spaces and more
20(7)
1.4.4 Derivatives in Banach spaces
27(5)
2 Elements of analysis for linear and nonlinear partial elliptic differential equations and systems
32(141)
2.1 Introduction
32(4)
2.2 Linear elliptic differential operators of second order, bilinear forms and solution concepts
36(9)
2.3 Bilinear forms and induced linear operators
45(9)
2.4 Linear elliptic differential operators, Fredholm alternative and regular solutions
54(23)
2.4.1 Introduction
54(4)
2.4.2 Linear operators of order 2m with C∞ coefficients
58(6)
2.4.3 Linear operators of order 2 under Cκ conditions
64(5)
2.4.4 Weak elliptic equation of order 2m in Hilbert spaces
69(8)
2.5 Nonlinear elliptic equations
77(36)
2.5.1 Introduction
77(2)
2.5.2 Definitions for nonlinear elliptic operators
79(2)
2.5.3 Special semilinear and quasilinear operators
81(7)
2.5.4 Quasilinear elliptic equations of order 2
88(8)
2.5.5 General nonlinear and Nemyckii operators
96(4)
2.5.6 Divergent quasilinear elliptic equations of order 2m
100(8)
2.5.7 Fully nonlinear elliptic equations of orders 2, m and 2m
108(5)
2.6 Linear and nonlinear elliptic systems
113(34)
2.6.1 Introduction
113(1)
2.6.2 General systems of elliptic differential equations
114(4)
2.6.3 Linear elliptic systems of order 2
118(7)
2.6.4 Quasilinear elliptic systems of order 2 and variational methods
125(7)
2.6.5 Linear elliptic systems of order 2m, m≥ 1
132(5)
2.6.6 Divergent quasilinear elliptic systems of order 2m
137(3)
2.6.7 Nemyekii operators and quasilinear divergent systems of order 2m
140(6)
2.6.8 Fully nonlinear elliptic systems of orders 2 and 2m
146(1)
2.7 Linearization of nonlinear operators
147(16)
2.7.1 Introduction
147(2)
2.7.2 Special semilinear and quasilinear equations
149(2)
2.7.3 Divergent quasilinear and fully nonlinear equations
151(6)
2.7.4 Quasilinear elliptic systems of orders 2 and 2m
157(1)
2.7.5 Linearizing general divergent quasilinear and fully nonlinear systems
158(5)
2.8 The Navier-Stokes equation
163(10)
2.8.1 Introduction
163(1)
2.8.2 The Stokes operator and saddle point problems
163(4)
2.8.3 The Navier-Stokes operator and its linearization
167(6)
PART II NUMERICAL METHODS
3 A general discretization theory
173(36)
3.1 Introduction
173(2)
3.2 Petrov-Galerkin and general discretization methods
175(10)
3.3 Variational and classical consistency
185(4)
3.4 Stability and consistency yield convergence
189(5)
3.5 Techniques for proving stability
194(9)
3.6 Stability implies invertibility
203(2)
3.7 Solving nonlinear systems: Continuation and Newton's method based upon the mesh independence principle (MIP)
205(4)
3.7.1 Continuation methods
205(1)
3.7.2 MIP for nonlinear systems
206(3)
4 Conforming finite element methods (FEMs)
209(87)
4.1 Introduction
209(3)
4.2 Approximation theory for finite elements
212(45)
4.2.1 Subdivisions and finite elements
212(2)
4.2.2 Polynomial finite elements, triangular and rectangular K
214(7)
4.2.3 Interpolation in finite element spaces, an example
221(8)
4.2.4 Interpolation errors and inverse estimates
229(4)
4.2.5 Inverse estimates on nonquasiuniform triangulations
233(5)
4.2.6 Smooth FEs on polyhedral domains, with O. Davydov
238(12)
4.2.7 Curved boundaries
250(7)
4.3 FEMs for linear problems
257(16)
4.3.1 Finite element methods: a simple example, essential tools
258(6)
4.3.2 Finite element methods for general linear equations and systems of orders 2 and 2m
264(2)
4.3.3 General convergence theory for conforming FEMs
266(7)
4.4 Finite element methods for divergent quasilinear elliptic equations and systems
273(4)
4.5 General convergence theory for monotone and quasilinear operators
277(4)
4.6 Mixed FEMs for Navier-Stokes and saddle point equations
281(7)
4.6.1 Navier-Stokes and saddle point equations
281(1)
4.6.2 Mixed FEMs for Stokes and saddle point equations
282(4)
4.6.3 Mixed FEMs for the Navier-Stokes operator
286(2)
4.7 Variational methods for eigenvalue problems
288(8)
4.7.1 Introduction
288(1)
4.7.2 Theory for eigenvalue problems
289(3)
4.7.3 Different variational methods for eigenvalue problems
292(4)
5 Nonconforming finite element methods
296(124)
5.1 Introduction
296(2)
5.2 Finite element methods for fully nonlinear elliptic problems
298(47)
5.2.1 Introduction
298(1)
5.2.2 Main ideas and results for the new FEM: An extended summary
299(6)
5.2.3 Fully nonlinear and general quasilinear elliptic equations
305(3)
5.2.4 Existence and convergence for semiconforming FEMs
308(3)
5.2.5 Definition of nonconforming FEMs
311(6)
5.2.6 Consistency for nonconforming FEMs
317(2)
5.2.7 Stability for the linearized operator and convergence
319(13)
5.2.8 Discretization of equations and systems of order 2m
332(4)
5.2.9 Consistency, stability and convergence for m, q ≥ 1
336(5)
5.2.10 Numerical solution of the FE equations with Newton's methods
341(4)
5.3 FE and other methods for nonlinear boundary conditions
345(1)
5.4 Quadrature approximate FEMs
346(22)
5.4.1 Introduction
346(2)
5.4.2 Quadrature and cubature formulas
348(2)
5.4.3 Quadrature for second order linear problems
350(7)
5.4.4 Quadrature for second order fully nonlinear equations
357(4)
5.4.5 Quadrature FEMs for equations and systems of order 2m
361(6)
5.4.6 Two useful propositions
367(1)
5.5 Consistency, stability and convergence for FEMs with variational crimes
368(52)
5.5.1 Introduction
368(2)
5.5.2 Variational crimes for our standard example
370(10)
5.5.3 FEMs with crimes for linear and quasilinear problems
380(7)
5.5.4 Discrete coercivity and consistency
387(3)
5.5.5 High order quadrature on edges
390(2)
5.5.6 Violated boundary conditions
392(7)
5.5.7 Violated continuity
399(7)
5.5.8 Stability for nonconforming FEMs
406(5)
5.5.9 Convergence, quadrature and solution of FEMs with crimes
411(3)
5.5.10 Isoparametric FEMs
414(6)
6 Adaptive finite element methods
420(35)
W. Dorfler
6.1 Introduction
420(10)
6.1.1 The model problem
421(1)
6.1.2 Singular solutions
421(2)
6.1.3 A priori error bounds
423(2)
6.1.4 Necessity of nonuniform mesh refinement
425(1)
6.1.5 Optimal meshes - A heuristic argument
425(2)
6.1.6 Optimal meshes for 2D corner singularities
427(1)
6.1.7 The finite element method-Notation and requirements
428(2)
6.2 The residual error estimator for the Poisson problem
430(19)
6.2.1 Upper a posteriori bound
430(2)
6.2.2 Lower a posteriori bound
432(1)
6.2.3 The a posteriori error estimate
433(1)
6.2.4 The adaptive finite element method
434(2)
6.2.5 Stable refinement methods for triangulations in R2
436(2)
6.2.6 Convergence of the adaptive finite element method
438(4)
6.2.7 Optimality
442(5)
6.2.8 Other types of estimators
447(1)
6.2.9 hp finite element method
448(1)
6.3 Estimation of quantities of interest
449(6)
6.3.1 Quantities of interest
449(1)
6.3.2 Error estimates for point errors
449(2)
6.3.3 Optimal meshes-A heuristic argument
451(1)
6.3.4 The general approach
451(4)
7 Discontinuous Galerkin methods (DCGMs)
455(105)
V. Dolejsi
7.1 Introdcution
455(4)
7.2 The model problem
459(2)
7.3 Discretization of the problem
461(11)
7.3.1 Triangulations
461(1)
7.3.2 Broken Sobolev spaces
462(1)
7.3.3 Extended variational formulation of the problem
463(6)
7.3.4 Discretization
469(3)
7.4 General linear elliptic problems
472(2)
7.5 Semilinear and quasilinear elliptic problem
474(8)
7.5.1 Semilinear elliptic problems
474(1)
7.5.2 Variational formulation and discretization of the problem
475(2)
7.5.3 Quasilinear elliptic systems
477(1)
7.5.4 Discretization of the quasilinear systems
478(4)
7.6 DCGMs are general discretization methods
482(4)
7.7 Geometry of the mesh, error and inverse estimates
486(5)
7.7.1 Geometry of the mesh
487(1)
7.7.2 Inverse and interpolation error estimates
487(4)
7.8 Penalty norms and consistency of the Jσh
491(3)
7.9 Coercive linearized principal parts
494(9)
7.9.1 Coercivity of the original linearized principal parts
494(1)
7.9.2 Coercivity and boundedness in Vh for the Laplacian
495(4)
7.9.3 Coercivity and boundedness in Vh for the general linear and the semilinear case
499(3)
7.9.4 Vh-coercivity and boundedness for quasilinear problems
502(1)
7.10 Consistency results for the ch, bh, lh
503(4)
7.10.1 Consistency of the ch and bh
503(2)
7.10.2 Consistency of the lh
505(2)
7.11 Consistency propeties of the ah
507(20)
7.11.1 Consistency of the ah for the Laplacian
507(4)
7.11.2 Consistency of the ah for general linear problems
511(3)
7.11.3 Consistency of the semilinear ah
514(4)
7.11.4 Consistency of the quasilinear ah for systems
518(5)
7.11.5 Consistency of the quasilinear ah for the equations of Houston, Robson, Suli, and for systems
523(4)
7.12 Convergence for DCGMs
527(5)
7.13 Solving nonlinear equations in DCGMs
532(6)
7.13.1 Introduction
532(1)
7.13.2 Discretized linearized quasilinear system and differentiable consistency
532(6)
7.14 hp-variants of DCGM
538(8)
7.14.1 hp-finite element spaces
539(1)
7.14.2 hp-DCGMs
540(1)
7.14.3 hp-inverse and approximation error estimates
540(2)
7.14.4 Consistency and convergence of hp-DCGMs
542(4)
7.15 Numerical experiences
546(14)
7.15.1 Scalar quasilinear equation
546(8)
7.15.2 System of the steady compressible Navier-Stokes equations
554(6)
8 Finite difference methods
560(75)
8.1 Introduction
560(2)
8.2 Difference methods for simple examples, notation
562(4)
8.3 Discrete Sobolev spaces
566(6)
8.3.1 Notation and definitions
566(3)
8.3.2 Discrete Sobolev spaces
569(3)
8.4 General elliptic problems with Dirichlet conditions, and their differnce methods
572(16)
8.4.1 General elliptic problems
572(2)
8.4.2 Second order linear elliptic difference equations
574(7)
8.4.3 Symmetric difference methods
581(2)
8.4.4 Linear equations of order 2m
583(1)
8.4.5 Quasilinear elliptic equations of orders 2, and 2m
584(2)
8.4.6 Systems of linear and quasilinear elliptic equations
586(1)
8.4.7 Fully nonlinear elliptic equations and systems
587(1)
8.5 Convergence for difference methods
588(22)
8.5.1 Discretization concepts in discrete Sobolev spaces
589(2)
8.5.2 The operators Ph, Q'h
591(3)
8.5.3 Consistency for difference equations
594(6)
8.5.4 Vhb-coercivity for linear(ized) elliptic difference equations
600(4)
8.5.5 Stability and convergence for general elliptic differece equations
604(6)
8.6 Natural boundary value problems of order 2
610(12)
8.6.1 Analysis for natural boundary value problems
611(2)
8.6.2 Difference methods for natural boundary value problems
613(9)
8.7 Other difference methods on curved boundaries
622(4)
8.7.1 The Shortley-Weller-Collatz method for linear equations
623(3)
8.8 Asymptotic expansions, extrapolation, and defect corrections
626(7)
8.8.1 A difference method based on polynomial interpolation for linear, and semilinear equations
627(3)
8.8.2 Asymptotic expansions for other methods
630(3)
8.9 Numerical experiments for the von Karman equations
633(2)
C.S. Chien
9 Variational methods for wavelets
635(51)
S. Dahlke
9.1 Introduction
635(2)
9.2 The scope of problems
637(2)
9.3 Wavelet analysis
639(14)
9.3.1 The discrete wavelet transform
640(4)
9.3.2 Biorthogonal bases
644(2)
9.3.3 Wavelets and function spaces
646(1)
9.3.4 Wavelets on domains
647(5)
9.3.5 Evaluation of nonlinear functionals
652(1)
9.4 Stable discretizations and preconditioning
653(6)
9.5 Applications to elliptic equations
659(5)
9.6 Saddle point and (Navier-)Stokes equations
664(5)
9.6.1 Saddle point equations
664(2)
9.6.2 Navier-Stokes equations
666(3)
9.7 Adaptive wavelet methods
669(17)
T. Raasch
9.7.1 Nonlinear approximation with wavelet systems
672(3)
9.7.2 Wavelet matrix compression
675(3)
9.7.3 Adaptive wavelet Galerkin methods
678(2)
9.7.4 Adaptive descent iterations
680(3)
9.7.5 Nonlinear stationary problems
683(3)
Bibliography 686(47)
Index 733
Professor Klaus Boehmer took his PhD in Pure and Applied Mathematics in 1969 at the University of Karlsruhe, Germany. He then worked in various universities in Germany and the USA, before becoming full professor at Phillipps University, Marburg, Germany in 1980. He has been a visiting professor at universities in China, the USA and Canada. He retired in 2001.