Adaptive numerical solution of PDEs
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| Format: | Buch |
| Sprache: | English |
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Berlin ; Boston
de Gruyter
2012
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| LEADER | 00000nam a2200000 c 4500 | ||
|---|---|---|---|
| 001 | BV040301749 | ||
| 003 | DE-604 | ||
| 005 | 20221130 | ||
| 007 | t | ||
| 008 | 120709s2012 a||| |||| 00||| eng d | ||
| 020 | |a 9783110283105 |9 978-3-11-028310-5 | ||
| 020 | |a 9783110283112 |9 978-3-11-028311-2 | ||
| 035 | |a (OCoLC)853206708 | ||
| 035 | |a (DE-599)BVBBV040301749 | ||
| 040 | |a DE-604 |b ger |e rda | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-739 |a DE-188 |a DE-19 |a DE-20 |a DE-634 |a DE-83 |a DE-91G |a DE-862 | ||
| 082 | 0 | |a 515/.3533 | |
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| 084 | |a SK 540 |0 (DE-625)143245: |2 rvk | ||
| 084 | |a SK 920 |0 (DE-625)143272: |2 rvk | ||
| 084 | |a 65Mxx |2 msc | ||
| 084 | |a 65Nxx |2 msc | ||
| 084 | |a MAT 671f |2 stub | ||
| 100 | 1 | |a Deuflhard, Peter |d 1944-2019 |0 (DE-588)108205983 |4 aut | |
| 245 | 1 | 0 | |a Adaptive numerical solution of PDEs |c Peter Deuflhard ; Martin Weiser |
| 264 | 1 | |a Berlin ; Boston |b de Gruyter |c 2012 | |
| 300 | |a xi, 421 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references and index | ||
| 650 | 4 | |a Differential equations, Elliptic / Numerical solutions / Textbooks | |
| 650 | 4 | |a Differential equations, Parabolic / Numerical solutions / Textbooks | |
| 650 | 0 | 7 | |a Adaptives Verfahren |0 (DE-588)4310560-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | 1 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |D s |
| 689 | 0 | 2 | |a Adaptives Verfahren |0 (DE-588)4310560-9 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Weiser, Martin |d 1970- |0 (DE-588)123252040 |4 aut | |
| 856 | 4 | 2 | |m Digitalisierung UB Passau |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025156768&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-025156768 | ||
Datensatz im Suchindex
| DE-BY-862_location | 2000 |
|---|---|
| DE-BY-FWS_call_number | 2000/SK 920 D485 |
| DE-BY-FWS_katkey | 579696 |
| DE-BY-FWS_media_number | 083000513421 |
| DE-BY-TUM_call_number | 0303/MAT 671f 2014 L 913 0303/MAT 671f 2014 L 913+5 0303/MAT 671f 2014 L 913+3 0303/MAT 671f 2014 L 913+4 0303/MAT 671f 2014 L 913+2 |
| DE-BY-TUM_katkey | 1906572 |
| DE-BY-TUM_media_number | 040071475691 040080097874 040080097885 040080097909 040080097896 |
| _version_ | 1816713840920887296 |
| adam_text | Contents
Preface
v
Outline 1
1
Elementary Partial Differential Equations
5
1.1
Laplace and
Poisson
Equation
................................ 5
1.1.1
Boundary Value Problems
............................ 6
1.1.2
Initial Value Problem
................................ 10
1.1.3
Eigenvalue Problem
................................. 12
1.2
Diffusion Equation
........................................ 15
1.3
Wave Equation
........................................... 18
1.4 Schrödinger
Equation
...................................... 23
1.5
Helmholtz Equation
........................................ 26
1.5.1
Boundary Value Problems
............................ 26
1.5.2
Time-harmonic Differential Equations
................... 27
1.6
Classification
............................................. 29
1.7
Exercises
................................................ 31
2
Partial Differential Equations in Science and Technology
34
2.1
Electrodynamics
.......................................... 34
2.1.1
Maxwell Equations
.................................. 34
2.1.2
Optical Model Hierarchy
.............................. 37
2.2
Fluid Dynamics
........................................... 40
2.2.1
Euler
Equations
..................................... 41
2.2.2
Navier-Stokes Equations
............................. 44
2.2.3
Prandtl s Boundary Layer
............................. 49
2.2.4
Porous Media Equation
............................... 51
2.3
Elastomechanics
.......................................... 52
2.3.1
Basic Concepts of Nonlinear Elastomechanics
............. 52
2.3.2
Linear Elastomechanics
.............................. 56
2.4
Exercises
................................................ 59
3
Finite
Difference Methods for
Poisson
Problems
62
3.1
Discretization of Standard Problem
............................ 62
3.1.1
Discrete Boundary Value Problems
..................... 63
3.1.2
Discrete Eigenvalue Problem
.......................... 68
3.2
Approximation Theory on Uniform Grids
....................... 71
3.2.1
Discretization Error in L2
............................. 73
3.2.2
Discretization Error in L°°
............................ 76
3.3
Discretization on
Nonuniform
Grids
........................... 78
3.3.1
One-dimensional Special Case
......................... 78
3.3.2
Curved Boundaries
.................................. 80
3.4
Exercises
................................................ 83
4
Galerkin Methods
86
4.1
General Scheme
........................................... 86
4.1.1
Weak Solutions
..................................... 86
4.1.2 Ritz
Minimization for Boundary Value Problems
.......... 89
4.1.3
Rayleigh-Ritz Minimization for Eigenvalue Problems
...... 93
4.2
Spectral Methods
.......................................... 95
4.2.1
Realization by Orthogonal Systems
..................... 96
4.2.2
Approximation Theory
............................... 100
4.2.3
Adaptive Spectral Methods
............................ 103
4.3
Finite Element Methods
.................................... 108
4.3.1
Meshes and Finite Element Spaces
...................... 108
4.3.2
Elementary Finite Elements
...........................
Ill
4.3.3
Realization of Finite Elements
......................... 121
4.4
Approximation Theory for Finite Elements
..................... 128
4.4.1
Boundary Value Problems
............................ 128
4.4.2
Eigenvalue Problems
................................. 131
4.4.3
Angle Condition for
Nonuniform
Meshes
................ 136
4.5
Exercises
................................................ 139
5
Numerical Solution of Linear Elliptic Grid Equations
143
5.1
Direct Elimination Methods
................................. 144
5.1.1
Symbolic Factorization
............................... 145
5.1.2
Frontal Solvers
..................................... 147
5.2
Matrix Decomposition Methods
.............................. 150
5.2.1
Jacobi Method
...................................... 152
5.2.2
Gauss-Seidel Method
................................ 154
5.3
Conjugate
Gradient
Method
.................................156
5.3.1
CG-Method as Galerkin Method
........................156
5.3.2
Preconditioning
.....................................159
5.3.3
Adaptive PCG-method
...............................163
5.3.4
A CG-variant for Eigenvalue Problems
..................165
5.4
Smoothing Property of Iterative Solvers
........................170
5.4.1
Illustration for the
Poisson
Model Problem
...............170
5.4.2
Spectral Analysis for Jacobi Method
....................174
5.4.3
Smoothing Theorems
................................175
5.5
Iterative Hierarchical Solvers
................................180
5.5.1
Classical Multigrid Methods
...........................182
5.5.2
Hierarchical-basis Method
............................190
5.5.3
Comparison with Direct Hierarchical Solvers
.............193
5.6
Power Optimization of a Darrieus Wind Generator
...............194
5.7
Exercises
................................................200
Construction of Adaptive Hierarchical Meshes
203
6.1
A Posteriori Error Estimators
................................203
6.1.1
Residual Based Error Estimator
........................206
6.1.2
Triangle Oriented Error Estimators
......................211
6.1.3
Gradient Recovery
..................................215
6.1.4
Hierarchical Error Estimators
..........................219
6.1.5
Goal-oriented Error Estimation
.........................222
6.2
Adaptive Mesh Refinement
..................................223
6.2.1
Equilibration of Local Discretization Errors
...............224
6.2.2
Refinement Strategies
................................229
6.2.3
Choice of Solvers on Adaptive Hierarchical Meshes
........233
6.3
Convergence on Adaptive Meshes
............................233
6.3.1
A Convergence Proof
................................234
6.3.2
An Example with a Reentrant Corner
....................236
6.4
Design of a Plasmon-Polariton Waveguide
......................240
6.5
Exercises
................................................244
Adaptive Multigrid Methods for Linear Elliptic Problems
246
7.1
Subspace Correction Methods
................................246
7.1.1
Basic Principle
.....................................247
7.1.2
Sequential Subspace Correction Methods
.................250
7.1.3
Parallel Subspace Correction Methods
...................255
7.1.4
Overlapping Domain Decomposition Methods
.............259
7.1.5
Higher-order Finite Elements
..........................266
7.2 Hierarchical Space
Decompositions
...........................271
7.2.1
Decomposition into Hierarchical Bases
..................272
7.2.2
L2-orthogonal Decomposition: BPX
....................278
7.3
Multigrid Methods Revisited
................................. 282
7.3.1
Additive Multigrid Methods
........................... 282
7.3.2
Multiplicative Multigrid Methods
....................... 286
7.4
Cascadic Multigrid Methods
................................. 289
7.4.1
Theoretical Derivation
............................... 289
7.4.2
Adaptive Realization
................................. 295
7.5
Eigenvalue Problem Solvers
................................. 300
7.5.1
Linear Multigrid Method
.............................. 301
7.5.2
Rayleigh Quotient Multigrid Method
.................... 303
7.6
Exercises
................................................ 306
Adaptive Solution of Nonlinear Elliptic Problems
310
8.1
Discrete Newton Methods for Nonlinear Grid Equations
........... 311
8.1.1
Exact Newton Methods
............................... 312
8.1.2
Inexact Newton-PCG Methods
......................... 316
8.2
Inexact Newton-Multigrid Methods
........................... 319
8.2.1
Hierarchical Grid Equations
........................... 319
8.2.2
Realization of Adaptive Algorithm
...................... 321
8.2.3
An Elliptic Problem Without a Solution
.................. 325
8.3
Operation Planning in Facial Surgery
.......................... 328
8.4
Exercises
................................................ 331
Adaptive Integration of Parabolic Problems
333
9.1
Time Discretization of Stiff Differential Equations
............... 333
9.1.1
Linear Stability Theory
............................... 334
9.1.2
Linearly Implicit One-step Methods
..................... 340
9.1.3
Order Reduction
.................................... 347
9.2
Space-time Discretization of Parabolic PDEs
.................... 353
9.2.1
Adaptive Method of Lines
............................ 354
9.2.2
Adaptive Method of Time Layers
....................... 362
9.2.3
Goal-oriented Error Estimation
......................... 371
9.3
Electrical Excitation of the Heart Muscle
....................... 374
9.3.1
Mathematical Models
................................ 374
9.3.2
Numerical Simulation
................................ 375
9.4
Exercises
................................................ 378
A Appendix
380
A.I Fourier Analysis and Fourier Transform
........................380
A.2 Differential Operators in K3
.................................381
A.3 Integral Theorems
.........................................383
A.4 Delta-Distribution and Green s Functions
.......................387
A.5 Sobolev Spaces
...........................................392
A.6 Optimality Conditions
......................................397
В
Software
398
B.I Adaptive Finite Element Codes
...............................398
B.2 Direct Solvers
............................................399
B.3 Nonlinear Solvers
.........................................399
Bibliography
401
Index
415
|
| any_adam_object | 1 |
| author | Deuflhard, Peter 1944-2019 Weiser, Martin 1970- |
| author_GND | (DE-588)108205983 (DE-588)123252040 |
| author_facet | Deuflhard, Peter 1944-2019 Weiser, Martin 1970- |
| author_role | aut aut |
| author_sort | Deuflhard, Peter 1944-2019 |
| author_variant | p d pd m w mw |
| building | Verbundindex |
| bvnumber | BV040301749 |
| classification_rvk | SK 500 SK 540 SK 920 |
| classification_tum | MAT 671f |
| ctrlnum | (OCoLC)853206708 (DE-599)BVBBV040301749 |
| dewey-full | 515/.3533 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.3533 |
| dewey-search | 515/.3533 |
| dewey-sort | 3515 43533 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV040301749 |
| illustrated | Illustrated |
| indexdate | 2024-11-25T17:37:10Z |
| institution | BVB |
| isbn | 9783110283105 9783110283112 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025156768 |
| oclc_num | 853206708 |
| open_access_boolean | |
| owner | DE-739 DE-188 DE-19 DE-BY-UBM DE-20 DE-634 DE-83 DE-91G DE-BY-TUM DE-862 DE-BY-FWS |
| owner_facet | DE-739 DE-188 DE-19 DE-BY-UBM DE-20 DE-634 DE-83 DE-91G DE-BY-TUM DE-862 DE-BY-FWS |
| physical | xi, 421 Seiten Illustrationen, Diagramme |
| publishDate | 2012 |
| publishDateSearch | 2012 |
| publishDateSort | 2012 |
| publisher | de Gruyter |
| record_format | marc |
| spellingShingle | Deuflhard, Peter 1944-2019 Weiser, Martin 1970- Adaptive numerical solution of PDEs Differential equations, Elliptic / Numerical solutions / Textbooks Differential equations, Parabolic / Numerical solutions / Textbooks Adaptives Verfahren (DE-588)4310560-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4310560-9 (DE-588)4128130-5 (DE-588)4044779-0 |
| title | Adaptive numerical solution of PDEs |
| title_auth | Adaptive numerical solution of PDEs |
| title_exact_search | Adaptive numerical solution of PDEs |
| title_full | Adaptive numerical solution of PDEs Peter Deuflhard ; Martin Weiser |
| title_fullStr | Adaptive numerical solution of PDEs Peter Deuflhard ; Martin Weiser |
| title_full_unstemmed | Adaptive numerical solution of PDEs Peter Deuflhard ; Martin Weiser |
| title_short | Adaptive numerical solution of PDEs |
| title_sort | adaptive numerical solution of pdes |
| topic | Differential equations, Elliptic / Numerical solutions / Textbooks Differential equations, Parabolic / Numerical solutions / Textbooks Adaptives Verfahren (DE-588)4310560-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Differential equations, Elliptic / Numerical solutions / Textbooks Differential equations, Parabolic / Numerical solutions / Textbooks Adaptives Verfahren Numerisches Verfahren Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025156768&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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