Numerical solution of elliptic and parabolic partial differential equations:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2013
|
| Edition: | 1. publ. |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIX, 635 S. graph. Darst. CD-ROM (12 cm) |
| ISBN: | 9780521877268 9781107043831 9781107688070 |
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MARC
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| 100 | 1 | |a Trangenstein, John A. |d 1949- |e Verfasser |0 (DE-588)111651735 |4 aut | |
| 245 | 1 | 0 | |a Numerical solution of elliptic and parabolic partial differential equations |c John A. Trangenstein |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2013 | |
| 300 | |a XIX, 635 S. |b graph. Darst. |e CD-ROM (12 cm) | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
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|---|---|
| adam_text | Contents
Preface
page
xv
1
Introduction to Partial Differential Equations
1
1.1
Types of Second-Order PDEs
1
1.2
Physical Problems
2
1.2.1
Heat Flow
3
1.2.2
Convection-Diffusion Equation
4
1.2.3
Electrocardiology
5
1.2.4
Miscible
Displacement
5
1.2.5
Thin Films
7
1.2.6
Incompressible Fluids
8
1.2.7
Elastic Solids
8
1.3
Summary
10
2
Parabolic Equations
13
2.1
Theory of Linear Parabolic Equations
13
2.1.1
Continuous Dependence on the Data
14
2.1.2
Green s Function
15
2.1.3
Reflection and Superposition
17
2.1.4
Maximum Principle
19
2.1.5
Bounded Domains and Eigenfunction Expansions
19
2.2
Finite Difference Methods in One Dimension
21
2.2.1
Continuous-In-Time Methods
21
2.2.2
Explicit Centered Differences
28
2.2.3
Implicit Centered Differences
38
2.2.4
Crank-Nicolson Scheme
43
2.2.5
Classical Higher-Order Temporal Discretization
48
2.2.6
Deferred Correction
51
2.3
Lax Convergence Theorem
59
2.4
Fourier Analysis
60
2.4.1
Constant-Coefficient Equations
60
vu
viii
Contents
2.4.2
Diffusion Problems
63
2.4.3
Applications
65
2.5
Lax
Equivalence Theorem
70
2.6
Measuring Accuracy and Efficiency
74
2.7
Finite Difference Methods in Multiple Dimensions
78
2.7.1
Unsplit Methods
78
2.7.2
Operator Splitting
80
Iterative Linear Algebra
85
3.1
Relative Efficiency of Implicit Computations
85
3.2
Vector Norms
89
3.3
Matrix Norms
90
3.4
Neumann Series
94
3.5
Perron-Frobenius Theorem
96
3.6
M
-Matrices
98
3.7
Iterative Improvement
103
3.7.1
Richardson^s Iteration
105
3.7.2
Jacobi Iteration
107
3.7.3
Gauss-Seidel Iteration
113
3.7.4
Successive Over-Relaxation
117
3.7.5
Termination Criteria for Iterative Methods
119
3.8
Gradient Methods
123
3.8.1
Steepest Descent
124
3.8.2
Conjugate Gradients
126
3.8.3
Preconditioned Conjugate Gradients
133
3.8.4
Biconjugate Gradients
135
3.9
Minimum Residual Methods
142
3.9.1
Orthomin
142
3.9.2
GMRES
146
3.10
Nonlinear Systems
153
3.10.1
Newton Algorithms
153
3.10.2
Nonlinear Krylov Algorithms
156
3.10.3
Nonlinear Case Study
157
3.11
Multigrid
158
3.11.1
V-Cycle
159
3.11.2
Projection
160
3.11.3
W-Cycle
162
3.11.4
Convergence
163
3.11.5
Condition Number
165
3.11.6
Prolongation
168
Contents ix
3.11.7 Multigrid
Debugging Techniques
176
4
Introduction to Finite Element Methods
179
4.1
Weak Formulation
179
4.2
Applications
183
4.2.1
Steady-State Heat Flow
183
4.2.2
Incompressible Single-Phase Flow in Porous Media
184
4.2.3
Linear Elasticity
184
4.2.4
Electromagnetism
187
4.3
Galerkin Methods
188
4.4
Finite Element Example
190
4.4.1
Nodal Viewpoint
191
4.4.2
Element Viewpoint
194
4.4.3
Finite Differences
196
4.5
Overview of Finite Elements
200
4.6
Reference Shapes
202
4.6.1
Intervals
203
4.6.2
Triangles
204
4.6.3
Quadrilaterals
205
4.6.4
Tetrahedra
206
4.6.5
Prisms
208
4.6.6
Hexahedra
209
4.7
Polynomial Families
210
4.7.1 Lagrange
Polynomials
211
4.7.2
Legendre Polynomials
215
4.7.3
Hierarchical Polynomials
216
4.8
Multi-Indices
217
4.9
Shape Function Families
219
4.9.1 Lagrange
Shape Functions
219
4.9.2
Hierarchical Shape Functions
221
4.10
Quadrature Rules
225
4.10.1
Newton-Cotes Quadrature
225
4.10.2
Clenshaw-Curtis Quadrature
226
4.10.3
Gaussian Quadrature
227
4.10.4
Lobatto Quadrature
230
4.10.5
Tensor Product Quadrature
232
4.10.6
Integrals in Barycentric Coordinates
232
4.10.7
Triangles
233
4.10.8
Quadratures on Tetrahedra
236
4.11
Mesh Generation
237
Contents
4.
12
Coordinate Mappings
4.12.1
Boundary Charts
4.12.2
Intervals
4.12.3
Quadrilaterals
4.12.4
Triangles
4.12.5
Hexahedra
4.12.6
Tetrahedra
4.12.7
Prisms
4.12.8
Continuity
4.
13
Finite Elements
4.
14
Linear Systems
4.14.1
Inhomogeneity Integrals
4.14.2
Differential Operator Integrals
4.14.3
Neumann Boundary Conditions
4.14.4
Dirichlet Boundary Conditions
4.14.5
Linear System Assembly
5
Finite
Element
Theory
5.
1
Norms
;
ind
Derivatives
5.1.1
Function Norms
5.1.2
Function Spaces
5.1.3
Differentiation
5.
2
Sobolev
Spaces
5.2.1
Sobolev Norms
5.2.2
Imbedding Theorems
5.2.3
Hubert Scales
5.2.4
Extension Theorem
5.2.5
Trace Theorems
5.2.6
Poincaré
Inequality
5.2.7
Friedrichs
Inequality
5
.3
Elliptic
Equations
5.3.1
Elliptic Differential Operators
5.3.2
Green s Formula
5.3.3
Dirichlet Problems
5
.4
Elliptic
Regularity
5.4.1
Coercivity
5.4.2
Well-Posedness
5.4.3
Garding s Inequality
5.4.4
Higher-Order Regularity
5.4.5
Linear Elasticity
238
239
241
241
242
247
248
251
251
253
253
255
256
257
260
261
263
263
264
265
268
274
275
279
281
284
284
287
288
289
289
292
295
300
300
302
305
307
311
Contents xi
5.5 Galerkin
Methods
313
5.5.1
Assumptions
314
5.5.2 Well-Posedness 316
5.5.3 Wf
Error Estimates
317
5.5.4
Convergence for Rough
Inhomogeneities 317
5.5.5
H° Error Estimates
318
5.5.6
Negative Norm Estimates
322
5.5.7
Non-Coercive Weak Forms
325
5.5.8
Max Norm Error Estimates
327
332
332
333
336
337
340
342
342
343
344
351
366
367
368
372
375
382
385
386
388
389
390
391
y
393
7
Mixed and Hybrid Finite Elements
398
7.1
HJ andH r
399
7.2
Physical Problems
401
7.2.1
Porous Flow
402
7.2.2
Stokes Equation
405
Finite Element
Approximations
6.1
Gaps
і
η
Our Theory
6.2
Finite
Element Assumptions
6.3
Piecewise Polynomial Approximation
6.3.1
Bramble-Hilbert Lemma
6.3.2
Lagrange
Polynomial Interpolation
6.3.3
Approximation versus Interpolation
6.4
Conforming Spaces
6.4.1
Sufficient Conditions
6.4.2
Linear Maps
6.4.3
Nonlinear Maps
6.5
Useful
Approximations
6.5.1
Strang
Lemma
6.5.2
Approximation Assumptions
6.5.3
Domain Approximation
6.5.4
Inhomogeneity Integrals
6.5.5
Stiffness Matrix Integrals
6.5.6
Summary
6.6
Refinement
6.7
Inverse
;
Estimates
6.8
Condition Number Estimates
6.8.1
A Posteriori Estimates
6.8.2
A Priori Estimates
6.8.3
Maximum Attainable Accuracy
xii
Contents
7.2.3
Linear Elasticity
407
7.2.4
Maxwell s Equations
410
7.3
Saddle-Point Problems
416
7.3.1
Quadratic Programming
416
7.3.2
Functional Analysis
421
7.3.3
Well-Posedness
426
7.4
Mixed Finite Elements
431
7.4.1
Porous Flow Example
431
7.4.2
Conforming Spaces
434
7.4.3
Coordinate Mappings
434
7.4.4
Linear Functionals
438
7.4.5
Mapped Linear Functionals
440
7.4.6
Interpolants
455
7.4.7
Interpolant
Divergence
457
7.4.8
Interpolation Errors
458
7.4.9
Inf-Sup Condition
460
7.4.10
Error Estimates
460
7.4.11
Standard Hdiv Discussion
463
7.4.12
Я
Conforming Conditions
466
7.4.13
Raviart-Thomas Spaces
467
7.4.14
Brezzi-Douglas-Marini Spaces
477
7.4.15
Brezzi-Douglas-Fortin-Marini Spaces
486
7.4.16
Linear Elasticity Spaces
490
7.4.17
Standard
Hcurl
Discussion
490
7.4.18
First
Nédélec
Spaces
493
7.4.19
Second
Nédélec
Spaces
500
7.5
Iterative Methods
505
7.5.1
Richardson Iteration
506
7.5.2
Steepest Descent
506
7.5.3
Conjugate Gradients
507
7.5.4
Penalty Methods
508
7.6
Hybrid Mixed Finite Elements
514
7.6.1
Problem Formulation
514
7.6.2
Method Formulation
516
7.6.3
Porous Flow Example
518
8
Finite Elements for Parabolic Equations
520
8.1
Well-Posedness
520
8.1.1
Existence
520
8.1.2
Continuous Dependence on the Data
523
Cont
en is
Xlii
8.2
Galerkin Methods
528
8.2.1
Spatial Discretization
529
8.2.2
Existence
530
8.2.3
Continuous Dependence on the Data
531
8.2.4
Time-Independent Elliptic Operator
532
8.2.5
Time-Dependent Elliptic Operator
534
8.2.6
Examples
535
8.3
Convection-Diffusion Problems
541
8.4
Reaction-Diffusion Problems
552
9
Finite Elements and Multigrid
554
9.1
Assumptions
554
9.2
Prolongation and Restriction
555
9.3
Coarse Grid Projection
562
9.4
Parabolic Problems
562
9.5
Mixed Methods
563
10
Local Refinement
564
10.1
Locally Refined Tessellations
564
10.2
Clement s Interpolation
567
10.3
Bubble Functions
570
10.4
Residual Estimator
574
10.4.1
Local Estimator
575
10.4.2
Efficiency
577
10.5
Other Error Estimators
579
10.6
Adaptive Mesh Refinement
580
10.7
Mortar Methods
581
10.7.1
Sub-Domains
581
10.7.2
Meshing
584
10.7.3
Finite Elements
585
10.7.4
Multiplier Spaces
587
10.7.5
Solution Spaces
602
Nomenclature
610
References
616
Author index
628
Subject index
631
|
| any_adam_object | 1 |
| author | Trangenstein, John A. 1949- |
| author_GND | (DE-588)111651735 |
| author_facet | Trangenstein, John A. 1949- |
| author_role | aut |
| author_sort | Trangenstein, John A. 1949- |
| author_variant | j a t ja jat |
| building | Verbundindex |
| bvnumber | BV041155229 |
| classification_rvk | SK 540 SK 910 SK 920 |
| ctrlnum | (OCoLC)852630870 (DE-599)BVBBV041155229 |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV041155229 |
| illustrated | Illustrated |
| indexdate | 2024-07-10T00:40:51Z |
| institution | BVB |
| isbn | 9780521877268 9781107043831 9781107688070 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026130621 |
| oclc_num | 852630870 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-20 DE-703 DE-634 |
| owner_facet | DE-355 DE-BY-UBR DE-20 DE-703 DE-634 |
| physical | XIX, 635 S. graph. Darst. CD-ROM (12 cm) |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| spelling | Trangenstein, John A. 1949- Verfasser (DE-588)111651735 aut Numerical solution of elliptic and parabolic partial differential equations John A. Trangenstein 1. publ. Cambridge [u.a.] Cambridge Univ. Press 2013 XIX, 635 S. graph. Darst. CD-ROM (12 cm) txt rdacontent n rdamedia nc rdacarrier Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Parabolische Differentialgleichung (DE-588)4173245-5 s Elliptische Differentialgleichung (DE-588)4014485-9 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Finite-Elemente-Methode (DE-588)4017233-8 s Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026130621&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Trangenstein, John A. 1949- Numerical solution of elliptic and parabolic partial differential equations Elliptische Differentialgleichung (DE-588)4014485-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd |
| subject_GND | (DE-588)4014485-9 (DE-588)4128130-5 (DE-588)4017233-8 (DE-588)4173245-5 |
| title | Numerical solution of elliptic and parabolic partial differential equations |
| title_auth | Numerical solution of elliptic and parabolic partial differential equations |
| title_exact_search | Numerical solution of elliptic and parabolic partial differential equations |
| title_full | Numerical solution of elliptic and parabolic partial differential equations John A. Trangenstein |
| title_fullStr | Numerical solution of elliptic and parabolic partial differential equations John A. Trangenstein |
| title_full_unstemmed | Numerical solution of elliptic and parabolic partial differential equations John A. Trangenstein |
| title_short | Numerical solution of elliptic and parabolic partial differential equations |
| title_sort | numerical solution of elliptic and parabolic partial differential equations |
| topic | Elliptische Differentialgleichung (DE-588)4014485-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd Parabolische Differentialgleichung (DE-588)4173245-5 gnd |
| topic_facet | Elliptische Differentialgleichung Numerisches Verfahren Finite-Elemente-Methode Parabolische Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026130621&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT trangensteinjohna numericalsolutionofellipticandparabolicpartialdifferentialequations |