Numerical methods for nonlinear elliptic differential equations: a synopsis
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Oxford [u.a.]
Oxford Univ. Press
2010
|
| Edition: | 1. publ. |
| Series: | Numerical mathematics and scientific computation
Oxford science publications |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXVII, 746 S. Ill., graph. Darst. |
| ISBN: | 9780199577040 0199577048 |
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| 100 | 1 | |a Böhmer, Klaus |d 1936-2020 |e Verfasser |0 (DE-588)142826197 |4 aut | |
| 245 | 1 | 0 | |a Numerical methods for nonlinear elliptic differential equations |b a synopsis |c Klaus Böhmer |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Oxford [u.a.] |b Oxford Univ. Press |c 2010 | |
| 300 | |a XXVII, 746 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Numerical mathematics and scientific computation | |
| 490 | 0 | |a Oxford science publications | |
| 650 | 4 | |a Differential equations, Elliptic |x Numerical solutions | |
| 650 | 4 | |a Differential equations, Nonlinear |x Numerical solutions | |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
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Record in the Search Index
| _version_ | 1804140886851846144 |
|---|---|
| adam_text | Contents
Preface
xiv
PART I ANALYTICAL RESULTS
1
From linear to nonlinear equations, fundamental results
3
1.1
Introduction
3
1.2
Linear versus nonlinear models
3
1.3
Examples for nonlinear partial differential equations
10
1.4
Fundamental results
13
1.4.1
Linear operators and functionals in Banach spaces
13
1.4.2
Inequalities and LP(Q) spaces
18
1.4.3
Holder and Sobolev spaces and more
20
1.4.4
Derivatives in Banach spaces
27
2
Elements of analysis for linear and nonlinear partial elliptic
differential equations and systems
32
2.1
Introduction
32
2.2
Linear elliptic differential operators of second order, bilinear forms
and solution concepts
36
2.3
Bilinear forms and induced linear operators
45
2.4
Linear elliptic differential operators,
Fredholm
alternative and regular
solutions
54
2.4.1
Introduction
54
2.4.2
Linear operators of order 2m with
C°°
coefficients
58
2.4.3
Linear operators of order
2
under Ck conditions
64
2.4.4
Weak elliptic equation of order 2m in Hubert spaces
69
2.5
Nonlinear elliptic equations
77
2.5.1
Introduction
77
2.5.2
Definitions for nonlinear elliptic operators
79
2.5.3
Special
semilinear
and
quasilinear
operators
81
2.5.4 Quasilinear
elliptic equations of order
2 88
2.5.5
General nonlinear and Nemyckii operators
96
2.5.6
Divergent
quasilinear
elliptic equations of order 2m
100
2.5.7
Fully nonlinear elliptic equations of orders
2,
m
and 2m
108
2.6
Linear and nonlinear elliptic systems
113
2.6.1
Introduction
113
2.6.2
General systems of elliptic differential equations
114
2.6.3
Linear elliptic systems of order
2 118
viii Contents
2.6.4 Quasilinear
elliptic systems of order
2
and
variational methods
125
2.6.5
Linear elliptic systems of order 2m,
m
> 1 132
2.6.6
Divergent
quasilinear
elliptic systems of order 2m
137
2.6.7
Nemyckii operators and
quasilinear
divergent systems
of order 2m 140
2.6.8
Fully nonlinear elliptic systems of orders
2
and 2m
146
2.7
Linearization of nonlinear operators
147
2.7.1
Introduction
147
2.7.2
Special
semilinear
and
quasilinear
equations
149
2.7.3
Divergent
quasilinear
and fully nonlinear equations
151
2.7.4 Quasilinear
elliptic systems of orders
2
and 2m
157
2.7.5
Linearizing general divergent
quasilinear
and fully
nonlinear systems
158
2.8
The Navier-Stokes equation
163
2.8.1
Introduction
163
2.8.2
The Stokes operator and saddle point problems
163
2.8.3
The Navier-Stokes operator and its linearization
167
PART
11
NUMERICAL METHODS
3
A general discretization theory
173
3.1
Introduction
173
3.2
Petrov-Galerkin and general discretization methods
175
3.3
Variational and classical consistency
185
3.4
Stability and consistency yield convergence
189
3.5
Techniques for proving stability
194
3.6
Stability implies invertibility
203
3.7
Solving nonlinear systems: Continuation and Newton s method
based upon the mesh independence principle
(МІР)
205
3.7.1
Continuation methods
205
3.7.2
МІР
for nonlinear systems
206
4
Conforming finite element methods (FEMs)
209
4.1
Introduction
209
4.2
Approximation theory for finite elements
212
4.2.1
Subdivisions and finite elements
212
4.2.2
Polynomial finite elements, triangular and
rectangular
К
214
4.2.3
Interpolation in finite element spaces, an example
221
4.2.4
Interpolation errors and inverse estimates
229
4.2.5
Inverse estimates on nonquasiuniform
triangulations
233
4.2.6
Smooth
FEs
on polyhedral domains, with O. Davydov
238
4.2.7
Curved boundaries
250
4.3
FEMs for linear problems
257
Contents ix
4.3.1
Finite element
methods: a simple example, essential tools
258
4.3.2
Finite element methods for general linear equations and
systems of orders
2
and 2m
264
4.3.3
General convergence theory for conforming FEMs
266
4.4
Finite element methods for divergent
quasilinear
elliptic
equations and systems
273
4.5
General convergence theory for monotone and
quasilinear
operators
277
4.6
Mixed FEMs for Navier-Stokes and saddle point equations
281
4.6.1
Navier-Stokes and saddle point equations
281
4.6.2
Mixed FEMs for Stokes and saddle point equations
282
4.6.3
Mixed FEMs for the Navier-Stokes operator
286
4.7
Variational methods for eigenvalue problems
288
4.7.1
Introduction
288
4.7.2
Theory for eigenvalue problems
289
4.7.3
Different variational methods for eigenvalue problems
292
Nonconforming finite element methods
296
5.1
Introduction
296
5.2
Finite element methods for fully nonlinear elliptic problems
298
5.2.1
Introduction
298
5.2.2
Main ideas and results for the new
FEM:
An extended summary
299
5.2.3
Fully nonlinear and general
quasilinear
elliptic equations
305
5.2.4
Existence and convergence for semiconforming FEMs
308
5.2.5
Definition of nonconforming FEMs
311
5.2.6
Consistency for nonconforming FEMs
317
5.2.7
Stability for the linearized operator and convergence
319
5.2.8
Discretization of equations and systems of order 2m
332
5.2.9
Consistency, stability and convergence for m,q
> 1 336
5.2.10
Numerical solution of the FE equations with
Newton s method
341
5.3
FE and other methods for nonlinear boundary conditions
345
5.4
Quadrature approximate FEMs
346
5.4.1
Introduction
346
5.4.2
Quadrature and
cubature
formulas
348
5.4.3
Quadrature for second order linear problems
350
5.4.4
Quadrature for second order fully nonlinear equations
357
5.4.5
Quadrature FEMs for equations and systems
of order 2m
361
5.4.6
Two useful propositions
367
5.5
Consistency, stability and convergence for FEMs with
variational crimes
368
5.5.1
Introduction
368
5.5.2
Variational crimes for our standard example
370
χ
Contents
5.5.3
FEMs with crimes for linear and
quasilinear
problems
380
5.5.4
Discrete coercivity and consistency
387
5.5.5
High order quadrature on edges
390
5.5.6
Violated boundary conditions
392
5.5.7
Violated continuity
399
5.5.8
Stability for nonconforming FEMs
406
5.5.9
Convergence, quadrature and solution of FEMs
with crimes
411
5.5.10
Isoparametric FEMs
414
6
Adaptive finite element methods, by W.
Dörfler 420
6.1
Introduction
420
6.1.1
The model problem
421
6.1.2
Singular solutions
421
6.1.3
A priori error bounds
423
6.1.4
Necessity of
nonuniform
mesh refinement
425
6.1.5
Optimal meshes
-
A heuristic argument
425
6.1.6
Optimal meshes for 2D corner singularities
427
6.1.7
The finite element method-Notation and
requirements
428
6.2
The residual error estimator for the
Poisson
problem
430
6.2.1
Upper a posteriori bound
430
6.2.2
Lower a posteriori bound
432
6.2.3
The a posteriori error estimate
433
6.2.4
The adaptive finite element method
434
6.2.5
Stable refinement methods for
triangulations
in K2
436
6.2.6
Convergence of the adaptive finite element method
438
6.2.7
Optimality
442
6.2.8
Other types of estimators
447
6.2.9
hp finite element method
448
6.3
Estimation of quantities of interest
449
6.3.1
Quantities of interest
449
6.3.2
Error estimates for point errors
449
6.3.3
Optimal meshes-A heuristic argument
451
6.3.4
The general approach
451
7
Discontinuous Galerkin methods (DCGMs), with
V. Dolejší
455
7.1
Introduction
455
7.2
The model problem
459
7.3
Discretization of the problem
461
7.3.1
Triangulations
461
7.3.2
Broken Sobolev spaces
462
7.3.3
Extended variational formulation of the problem
463
7.3.4
Discretization
459
7.4
General linear elliptic problems
472
Contents xi
7.5 Semilinear and quasilinear
elliptic problems
474
7.5.1 Semilinear
elliptic problems
474
7.5.2 Variational
formulation and discretization of
the problem
475
7.5.3 Quasilinear
elliptic systems
477
7.5.4
Discretization of the
quasilinear
systems
478
7.6
DCGMs are general discretization methods
482
7.7
Geometry of the mesh, error and inverse estimates
486
7.7.1
Geometry of the mesh
487
7.7.2
Inverse and interpolation error estimates
487
7.8
Penalty norms and consistency of the J£
491
7.9
Coercive linearized principal parts
494
7.9.1
Coercivity of the original linearized principal parts
494
7.9.2
Coercivity and boundedness in Vh for the Laplacian
495
7.9.3
Coercivity and boundedness in Vh for the general linear
and the
semilinear
case
499
7.9.4
V^-coercivity and boundedness for
quasilinear
problems
502
7.10
Consistency results for the ch,bh,
Ѓ
503
7.10.1
Consistency of the ch and bh
503
7.10.2
Consistency of the
Ѓ
505
7.11
Consistency properties of the
a/¡
507
7.11.1
Consistency of the a^ for the Laplacian
507
7.11.2
Consistency of the a^ for general linear problems
511
7.11.3
Consistency of the semilinear a^
514
7.11.4
Consistency of the
quasilinear
о^
for systems
518
7.11.5
Consistency of the
quasilinear
а^
for the equations of
Houston, Robson,
Süli,
and for systems
523
7.12
Convergence for DCGMs
527
7.13
Solving nonlinear equations in DCGMs
532
7.13.1
Introduction
532
7.13.2
Discretized linearized
quasilinear
system and
differentiable consistency
532
7.14
hp-variants of DCGM
538
7.14.1
hp-ňnite
element spaces
539
7.14.2
hp-OCGMs
540
7.14.3
/ip-inverse and approximation error estimates
540
7.14.4
Consistency and convergence of /ip-DCGMs
542
7.15
Numerical experiences
546
7.15.1
Scalar
quasilinear
equation
546
7.15.2
System of the steady compressible Navier-Stokes
equations
554
Finite difference methods
560
8.1
Introduction
560
8.2
Difference methods for simple examples, notation
562
xii Contents
8.3
Discrete Sobolev
spaces
566
8.3.1 Notation
and definitions
566
8.3.2
Discrete Sobolev spaces
569
8.4
General elliptic problems with Dirichlet conditions, and their
difference methods
572
8.4.1
General elliptic problems
572
8.4.2
Second order linear elliptic difference equations
574
8.4.3
Symmetric difference methods
581
8.4.4
Linear equations of order 2m
583
8.4.5 Quasilinear
elliptic equations of orders
2,
and 2m
584
8.4.6
Systems of linear and
quasilinear
elliptic equations
586
8.4.7
Fully nonlinear elliptic equations and systems
587
8.5
Convergence for difference methods
588
8.5.1
Discretization concepts in discrete Sobolev spaces
589
8.5.2
The operators Ph,Q h
591
8.5.3
Consistency for difference equations
594
8.5.4
V^-coercivity for linear(ized) elliptic difference equations
600
8.5.5
Stability and convergence for general elliptic difference
equations
604
8.6
Natural boundary value problems of order
2 610
8.6.1
Analysis for natural boundary value problems
611
8.6.2
Difference methods for natural boundary value
problems
613
8.7
Other difference methods on curved boundaries
622
8.7.1
The Shortley-Weller-Collatz method for linear equations
623
8.8
Asymptotic expansions, extrapolation, and defect corrections
626
8.8.1
A difference method based on polynomial interpolation for
linear, and
semilinear
equations
627
8.8.2
Asymptotic expansions for other methods
630
8.9
Numerical experiments for the
von Kármán
equations,
with C.S.
Chien
633
9
Variational methods for wavelets, with S. Dahlke
635
9.1
Introduction
635
9.2
The scope of problems
637
9.3
Wavelet analysis
639
9.3.1
The discrete wavelet transform
640
9.3.2 Biorthogonal
bases
644
9.3.3
Wavelets and function spaces
646
9.3.4
Wavelets on domains
647
9.3.5
Evaluation of nonlinear functionals
652
9.4
Stable discretizations and preconditioning
653
9.5
Applications to elliptic equations
659
9.6
Saddle point and (Navier-)Stokes equations
664
9.6.1
Saddle point equations
664
9.6.2
Navier-Stokes equations
666
Contents xiii
9.7
Adaptive
wavelet methods, by T. Raasch
669
9.7.1
Nonlinear approximation with wavelet systems
672
9.7.2
Wavelet matrix compression
675
9.7.3
Adaptive wavelet-Galerkin methods
678
9.7.4
Adaptive descent iterations
680
9.7.5
Nonlinear stationary problems
683
Bibliography
686
Index
733
|
| any_adam_object | 1 |
| author | Böhmer, Klaus 1936-2020 |
| author_GND | (DE-588)142826197 |
| author_facet | Böhmer, Klaus 1936-2020 |
| author_role | aut |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.3533 |
| dewey-search | 515.3533 |
| dewey-sort | 3515.3533 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV035905655 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T22:07:06Z |
| institution | BVB |
| isbn | 9780199577040 0199577048 |
| language | English |
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| physical | XXVII, 746 S. Ill., graph. Darst. |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | Oxford Univ. Press |
| record_format | marc |
| series2 | Numerical mathematics and scientific computation Oxford science publications |
| spelling | Böhmer, Klaus 1936-2020 Verfasser (DE-588)142826197 aut Numerical methods for nonlinear elliptic differential equations a synopsis Klaus Böhmer 1. publ. Oxford [u.a.] Oxford Univ. Press 2010 XXVII, 746 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Numerical mathematics and scientific computation Oxford science publications Differential equations, Elliptic Numerical solutions Differential equations, Nonlinear Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 gnd rswk-swf Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018763025&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Böhmer, Klaus 1936-2020 Numerical methods for nonlinear elliptic differential equations a synopsis Differential equations, Elliptic Numerical solutions Differential equations, Nonlinear Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 gnd |
| subject_GND | (DE-588)4128130-5 (DE-588)4310554-3 |
| title | Numerical methods for nonlinear elliptic differential equations a synopsis |
| title_auth | Numerical methods for nonlinear elliptic differential equations a synopsis |
| title_exact_search | Numerical methods for nonlinear elliptic differential equations a synopsis |
| title_full | Numerical methods for nonlinear elliptic differential equations a synopsis Klaus Böhmer |
| title_fullStr | Numerical methods for nonlinear elliptic differential equations a synopsis Klaus Böhmer |
| title_full_unstemmed | Numerical methods for nonlinear elliptic differential equations a synopsis Klaus Böhmer |
| title_short | Numerical methods for nonlinear elliptic differential equations |
| title_sort | numerical methods for nonlinear elliptic differential equations a synopsis |
| title_sub | a synopsis |
| topic | Differential equations, Elliptic Numerical solutions Differential equations, Nonlinear Numerical solutions Numerisches Verfahren (DE-588)4128130-5 gnd Nichtlineare elliptische Differentialgleichung (DE-588)4310554-3 gnd |
| topic_facet | Differential equations, Elliptic Numerical solutions Differential equations, Nonlinear Numerical solutions Numerisches Verfahren Nichtlineare elliptische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=018763025&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT bohmerklaus numericalmethodsfornonlinearellipticdifferentialequationsasynopsis |