Numerical models for differential problems:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Milan [u. a.]
Springer
2009
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| Schriftenreihe: | MS&A
2 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | Aus d. Ital. übers. |
| Beschreibung: | XVI, 601 S. Ill., graph. Darst. |
| ISBN: | 9788847010703 9788847010710 |
Internformat
MARC
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| 008 | 081215s2009 ad|| |||| 00||| eng d | ||
| 020 | |a 9788847010703 |9 978-88-470-1070-3 | ||
| 020 | |a 9788847010710 |9 978-88-470-1071-0 | ||
| 035 | |a (OCoLC)288986457 | ||
| 035 | |a (DE-599)BVBBV035213576 | ||
| 040 | |a DE-604 |b ger |e rakwb | ||
| 041 | 0 | |a eng | |
| 049 | |a DE-20 |a DE-355 |a DE-703 |a DE-83 |a DE-384 |a DE-29T |a DE-188 |a DE-11 | ||
| 050 | 0 | |a QA377 | |
| 082 | 0 | |a 518.64 |2 22 | |
| 084 | |a SK 540 |0 (DE-625)143245: |2 rvk | ||
| 084 | |a SK 910 |0 (DE-625)143270: |2 rvk | ||
| 084 | |a SK 920 |0 (DE-625)143272: |2 rvk | ||
| 084 | |a 65Mxx |2 msc | ||
| 084 | |a 65Nxx |2 msc | ||
| 100 | 1 | |a Quarteroni, Alfio |d 1952- |e Verfasser |0 (DE-588)120370158 |4 aut | |
| 240 | 1 | 0 | |a Modellistica numerica per problemi differenziali |
| 245 | 1 | 0 | |a Numerical models for differential problems |c Alfio Quarteroni |
| 264 | 1 | |a Milan [u. a.] |b Springer |c 2009 | |
| 300 | |a XVI, 601 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a MS&A |v 2 | |
| 500 | |a Aus d. Ital. übers. | ||
| 650 | 4 | |a Differential equations, Partial |x Numerical solutions | |
| 650 | 4 | |a Numerical analysis | |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Numerisches Verfahren |0 (DE-588)4128130-5 |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 | |5 DE-604 | |
| 830 | 0 | |a MS&A |v 2 |w (DE-604)BV035366143 |9 2 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017019770&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-017019770 | ||
Datensatz im Suchindex
| _version_ | 1804138472964882432 |
|---|---|
| adam_text | Contents
Preface
........................................................
V
1
A brief survey on partial differential equations
..................... 1
1.1
Definitions and examples
.................................... 1
1.2
Numerical solution
......................................... 3
1.3
PDE Classification
......................................... 5
1.3.1
Quadratic form associated to a PDE
................... 8
1.4
Exercises
................................................. 9
2
Elements of functional analysis
................................... 11
2.1
Functionals and bilinear forms
................................ 11
2.2
Differentiation in linear spaces
............................... 13
2.3
Elements of distributions
.................................... 15
2.3.1
Square-integrable functions
.......................... 17
2.3.2
Derivation in the sense of distributions
................. 18
2.4
Sobolev spaces
............................................ 20
2.4.1
Regularity of the
Η* (Ω)
spaces
....................... 21
2.4.2
The
Ha (ß)
space
................................... 22
2.4.3
Trace
operatore
.................................... 23
2.5
The spaces
L00
(ß) andLp(ß),
withl <p<oo
................. 24
2.6
Adjoint operators of a linear operator
.......................... 26
2.7
Spaces of time-dependent functions
........................... 27
2.8
Exercises
................................................. 28
3
Elliptic equations
............................................... 31
3.1
An elliptic problem example:
the
Poisson
equation
...................................... 31
3.2
The
Poisson
problem in the one-dimensional case
............... 32
3.2.1
Homogeneous Dirichlet problem
...................... 33
3.2.2
Non-homogeneous Dirichlet problem
.................. 39
3.2.3
Neumann Problem
.................................. 39
X
Contents
3.2.4
Mixed homogeneous
problem
........................ 40
3.2.5
Mixed (or Robin) boundary conditions
................. 40
3.3
The
Poisson
problem in the two-dimensional case
............... 41
3.3.1
The homogeneous Dirichlet problem
.................. 41
3.3.2
Equivalence, in the sense of distributions, between weak
and strong form of the Dirichlet problem
............... 43
3.3.3
The problem with mixed,
non
homogeneous conditions
... 44
3.3.4
Equivalence, in the sense of distributions, between weak
and strong form of the Neumann problem
.............. 46
3.4
More general elliptic problems
............................... 48
3.5
Existence and uniqueness theorem
............................ 50
3.6
Adjoint operator and adjoint problem
.......................... 51
3.7
Exercises
................................................. 56
4
The Galerkin finite element method for elliptic problems
............ 61
4.1
Approximation via the Galerkin method
....................... 61
4.2
Analysis of the Galerkin method
.............................. 63
4.2.1
Existence and uniqueness
........................... 63
4.2.2
Stability
.......................................... 64
4.2.3
Convergence
...................................... 64
4.3
The finite element method in the one-dimensional case
........... 66
4.3.1
The space
Χ
...................................... 67
4.3.2
The space X%
...................................... 68
4.3.3
The approximation with linear finite elements
........... 71
4.3.4
Interpolation operator and interpolation error
........... 73
4.3.5
Estimate of the finite element error in the H1 norm
....... 75
4.4
Finite elements, simplices and barycentric coordinates
........... 76
4.4.1
An abstract definition of finite element in the Lagrangian
case
.............................................. 76
4.4.2
Simplices
......................................... 78
4.4.3
Barycentric coordinates
............................. 78
4.5
The finite element method in the multi-dimensional case
.......... 80
4.5.1
Finite element solution of the
Poisson
problem
.......... 82
4.5.2
Conditioning of the stiffness matrix
................... 85
4.5.3
Estimate of the approximation error in the energy norm
... 88
4.5.4
Estimate of the approximation error in the L2 norm
...... 96
4.6
Grid adaptivity
............................................. 99
4.6.1
A priori adaptivity based on derivatives reconstruction
___ 100
4.6.2
A posteriori adaptivity
.............................. 103
4.6.3
Numerical examples of adaptivity
..................... 107
4.6.4
A posteriori error estimates in the L2 norm
............. 110
4.6.5
A posteriori estimates of a functional of the error
........ 112
4.7
Exercises
................................................. 114
Contents
XI
Parabolic
equations
............................................. 119
5.1
Weak formulation and its approximation
....................... 120
5.2
A priori estimates
.......................................... 123
5.3
Convergence analysis of the semi-discrete problem
.............. 126
5.4
Stability analysis of the 0-method
............................. 130
5.5
Convergence analysis of the ^-method
......................... 134
5.6
Exercises
................................................. 136
Generation of ID and 2D grids
................................... 139
6.1
Grid generation in ID
....................................... 139
6.2
Grid of a polygonal domain
.................................. 142
6.3
Generation of structured grids
................................ 144
6.4
Generation of non-structured grids
............................ 147
6.4.1
Delaunay
triangulation
.............................. 147
6.4.2
Advancing front technique
........................... 151
6.5
Regularization techniques
................................... 153
6.5.1
Diagonal exchange
................................. 154
6.5.2
Node displacement
................................. 155
Algorithms for the solution of linear systems
....................... 159
7.1
Direct methods
............................................ 159
7.2
Iterative methods
........................................... 162
7.2.1
Classical iterative methods
........................... 162
7.2.2
Gradient and conjugate gradient methods
............... 164
7.2.3
Krylov subspace methods
............................ 167
Elements of finite element programming
........................... 173
8.1
Operational phases of a finite element code
..................... 174
8.1.1
Code in a nutshell
.................................. 176
8.2
Numerical computation of integrals
........................... 177
8.2.1
Numerical integration using barycentric coordinates
..... 179
8.3
Storage of sparse matrices
................................... 182
8.4
Assembly phase
............................................ 186
8.4.1
Coding geometrical information
...................... 188
8.4.2
Coding of functional information
..................... 192
8.4.3
Mapping between reference and physical element
....... 193
8.4.4
Construction of local and global systems
............... 197
8.4.5
Boundary conditions prescription
..................... 201
8.5
Integration in time
.......................................... 204
8.6
A complete example
........................................ 207
The finite volume method
........................................ 217
9.1
Some basic principles
....................................... 218
9.2
Construction of control volumes for vertex-centered schemes
...... 220
9.3
Discretization of a diffusion-transport-reaction problem
.......... 223
XII Contents
9.4
Analysis of the finite volume approximation
.................... 225
9.5
Implementation of boundary conditions
........................ 226
10
Spectral methods
................................................ 227
10.1
The spectral Galerkin method for elliptic problems
.............. 227
10.2
Orthogonal polynomials and Gaussian numerical integration
...... 231
10.2.1
Orthogonal Legendre polynomials
.................... 231
10.2.2
Gaussian integration
................................ 234
10.2.3
Gauss-Legendre-Lobatto formulae
.................... 235
10.3
G-NI methods in one dimension
.............................. 237
10.3.1
Algebraic interpretation of the G-NI method
............ 239
10.3.2
Conditioning of the stiffness matrix in the G-NI method
.. 241
10.3.3
Equivalence between G-NI and collocation methods
..... 242
10.3.4
G-NI for parabolic equations
......................... 245
10.4
Generalization to the two-dimensional case
..................... 247
10.4.1
Convergence of the G-NI method
..................... 250
10.5
G-NI and SEM-NI methods for a one-dimensional model problem
. 257
10.5.1
TheG-NImethod
.................................. 257
10.5.2
The SEM-NI method
................................ 261
10.6
Spectral methods on triangles and tetrahedra
.................... 264
10.7
Exercises
................................................. 268
11
Diffusion-transport-reaction
equations
............................ 271
11.1
Weak problem formulation
................................... 271
11.2
Analysis of a one-dimensional diffusion-transport problem
........ 274
11.3
Analysis of a one-dimensional diffusion-reaction problem
........ 278
11.4
Finite elements and finite differences (FD)
..................... 280
11.5
The mass-lumping technique
................................. 281
11.6
Decentered FD schemes and artificial diffusion
.................. 284
11.7
Eigenvalues of the diffusion-transport equation
.................. 286
11.8
Stabilization methods
....................................... 289
11.8.1
Artificial diffusion and decentered finite element schemes
. 289
11.8.2
The Petrov-Galerkin method
......................... 292
11.8.3
The artificial diffusion and streamline-diffusion methods
in the two-dimensional case
.......................... 292
11.8.4
Consistence and truncation error for the Galerkin and
generalized Galerkin methods
........................ 294
11.8.5
Symmetric and skew-symmetric part of an operator
...... 295
11.8.6
Strongly consistent methods (GLS, SUPG)
............. 296
11.8.7
Analysis of the GLS method
......................... 298
11.8.8
Stabilization through bubble functions
................. 304
11.9
Some numerical tests
....................................... 306
11.10
An example of goal-oriented adaptivity
........................ 307
11.11
Exercises
................................................. 309
Contents
ХШ
12
Finite differences for hyperbolic equations
......................... 313
12.1
A scalar transport problem
................................... 313
12.1.1
An a priori estimate
................................. 315
12.2
Systems of linear hyperbolic equations
......................... 317
12.2.1
The wave equation
................................. 319
12.3
The finite difference method
................................. 320
12.3.1
Discretization of the scalar equation
................... 321
12.3.2
Discretization of linear hyperbolic systems
............. 323
12.3.3
Boundary treatment
................................. 324
12.4
Analysis of the finite difference methods
....................... 324
12.4.1
Consistency and convergence
......................... 324
12.4.2
Stability
.......................................... 325
12.4.3 Von
Neumann analysis and amplification coefficients
.... 330
12.4.4
Dissipation and dispersion
........................... 335
12.5
Equivalent equations
........................................ 337
12.5.1
The upwind scheme case
............................ 337
12.5.2
The Lax-Friedrichs and Lax-Wendroff case
............. 341
12.5.3
On the meaning of coefficients in equivalent equations
... 342
12.5.4
Equivalent equations and error analysis
................ 342
12.6
Exercises
................................................. 343
13
Finite elements and spectral methods for hyperbolic equations
....... 345
13.1
Temporal discretization
..................................... 345
13.1.1
The forward and backward
Euler
schemes
.............. 345
13.1.2
The upwind, Lax-Friedrichs and Lax-Wendroff schemes
.. 347
13.2
Taylor-Galerkin schemes
.................................... 350
13.3
The multi-dimensional case
.................................. 356
13.3.1
Semi-discretization: strong and weak treatment of the
boundary conditions
................................ 356
13.3.2
Temporal discretization
............................. 359
13.4
Discontinuous finite elements
................................ 362
13.4.1
The one-dimensional case
........................... 362
13.4.2
The multi-dimensional case
.......................... 367
13.5
Approximation using spectral methods
......................... 370
13.5.1
The G-NI method in a single interval
.................. 370
13.5.2
The DG-SEM-NI method
............................ 374
13.6
Numerical treatment of boundary conditions for hyperbolic systems
376
13.6.1
Weak treatment of boundary conditions
................ 379
13.7
Exercises
................................................. 382
14
Nonlinear hyperbolic problems
................................... 383
14.1
Scalar equations
............................................ 383
14.2
Finite difference approximation
............................... 388
14.3
Approximation by discontinuous finite elements
................. 389
14.4
Nonlinear hyperbolic systems
................................ 397
XIV Contents
IS Navier-Stokes equations
......................................... 401
15.1
Weak formulation of Navier-Stokes equations
................... 403
15.2
Stokes equations and their approximation
...................... 407
15.3
Saddle-point problems
...................................... 411
15.3.1
Problem formulation
................................ 411
15.3.2
Problem analysis
................................... 412
15.3.3
Galerkin approximation, stability and convergence analysis
416
15.4
Algebraic formulation of the Stokes problem
................... 420
15.5
An example of stabilized problem
............................. 424
15.6
A numerical example
....................................... 426
15.7
Time discretization of Navier-Stokes equations
.................. 427
15.7.1
Finite difference methods
............................ 429
15.7.2
Characteristics (or Lagrangian) methods
............... 430
15.7.3
Fractional step methods
............................. 431
15.8
Algebraic factorization methods and preconditioners for
saddle-point systems
........................................ 435
15.9
Free surface flow problems
.................................. 440
15.9.1
Navier-Stokes equations with variable density and
viscosity
.......................................... 441
15.9.2
Boundary conditions
................................ 443
15.9.3
Application to free surface flows
...................... 444
15.10
Interface evolution modeling
................................. 445
15.10.1
Explicit interface descriptions
........................ 445
15.10.2
Implicit interface descriptions
........................ 446
15.11
Finite volume approximation
................................. 450
15.12
Exercises
................................................. 453
16
Optimal control of partial differential equations
.................... 457
16.1
Definition of optimal control problems
......................... 457
16.2
A control problem for linear systems
.......................... 459
16.3
Some examples of optimal control problems for the Laplace
equation
.................................................. 460
16.4
On the minimization of linear functionals
...................... 461
16.5
The theory of optimal control for elliptic problems
............... 464
16.6
Some examples of optimal control problems
.................... 468
16.6.1
A Dirichlet problem with distributed control
............ 468
16.6.2
A Neumann problem with distributed control
........... 469
16.6.3
A Neumann problem with boundary control
............ 470
16.7
Numerical tests
............................................ 470
16.8
Lagrangian formulation of control problems
.................... 476
16.8.1
Constrained optimization in Rn
....................... 476
16.8.2
The solution approach based on the Lagrangian
......... 477
16.9
Iterative solution of the optimal control problem
................. 480
16.10
Numerical examples
........................................ 484
16.10.1
Heat dissipation by a thermal fin
...................... 485
Contents
XV
16.10.2
Thermal pollution in a river
.......................... 487
16.11
A
few considerations about observability and controllability
....... 489
16.12
Two alternative paradigms for numerical approximation
.......... 490
16.13
A numerical approximation of an optimal control problem for
advection-diffusion equations
................................ 492
16.13.1
The strategies optimize-then-discretize and
discretize-then-optimize
.......................... 494
16.13.2
A posteriori error estimates
.......................... 495
16.13.3
A test problem on control of pollutant emission
......... 497
16.14
Exercises
................................................. 499
17
Domain decomposition methods
.................................. 501
17.1
Three classical iterative DD methods
.......................... 502
17.1.1 Schwarz
method
................................... 502
17.1.2
Dirichlet-Neumann method
.......................... 504
17.1.3
Neumann-Neumann algorithm
........................ 506
17.1.4
Robin-Robin algorithm
.............................. 506
17.2
Multi-domain formulation of
Poisson
problem and interface
conditions
................................................. 507
17.2.1
The
Steklov-Poincaré
operator
........................ 507
17.2.2
Equivalence between Dirichlet-Neumann and Richardson
methods
.......................................... 509
17.3
Multidomain formulation of the finite element approximation of
the
Poisson
problem
........................................ 512
17.3.1
The
Schur
complement
.............................. 514
17.3.2
The discrete
Steklov-Poincaré
operator
................ 515
17.3.3
Equivalence between Dirichlet-Neumann and Richardson
methods in the discrete case
.......................... 518
17.4
Generalization to the case of many
subdomains
................. 519
17.4.1
Some numerical results
.............................. 521
17.5
DD preconditioners in case of many
subdomains
................ 523
17.5.1
Jacobi preconditioner
............................... 524
17.5.2
Bramble-Pasciak-Schatz preconditioner
................ 526
17.5.3
Neumann-Neumann preconditioner
................... 526
17.6 Schwarz
iterative methods
................................... 530
17.6.1
Algebraic form of
Schwarz
method for finite element
discretizations
..................................... 531
17.6.2 Schwarz
preconditioners
............................. 533
17.6.3
Two-level
Schwarz
preconditioners
.................... 536
17.7
An abstract convergence result
............................... 539
17.8
Interface conditions for other differential problems
.............. 540
17.9
Exercises
................................................. 544
XVI Contents
18
Reduced basis approximation for parametrized partial differential
equations
....................................................... 547
18.1
Elliptic coercive parametric PDEs
............................. 548
18.1.1
An illustrative example
.............................. 550
18.2
Geometric parametrization
................................... 551
18.2.1 Affine
geometry precondition
........................ 551
18.2.2 Affine
mappings: single
subdomain
................... 553
18.2.3
Piecewise
affine
mappings: multiple
subdomains
........ 556
18.2.4
Bilinearforms
..................................... 557
18.2.5
A second illustrative example
........................ 560
18.3
The reduced basis method
................................... 562
18.3.1
Reduced basis approximation and spaces
............... 562
18.3.2
Sampling strategies
................................. 566
18.4
Convergence of RB approximations
........................... 567
18.4.1
A priori convergence theory: single parameter case:
Ρ
= 1 567
18.4.2
Convergence:
Ρ
> 1................................ 568
18.5
A posteriori error estimation
................................. 572
18.5.1
Preliminaries
...................................... 573
18.5.2
Error bounds
...................................... 573
18.5.3
Offline-online computational procedure
................ 574
18.6
Historical perspective, background and extensions
............... 576
18.7
Exercises
................................................. 577
References
.......................................................... 581
Index
.............................................................. 595
|
| any_adam_object | 1 |
| author | Quarteroni, Alfio 1952- |
| author_GND | (DE-588)120370158 |
| author_facet | Quarteroni, Alfio 1952- |
| author_role | aut |
| author_sort | Quarteroni, Alfio 1952- |
| author_variant | a q aq |
| building | Verbundindex |
| bvnumber | BV035213576 |
| callnumber-first | Q - Science |
| callnumber-label | QA377 |
| callnumber-raw | QA377 |
| callnumber-search | QA377 |
| callnumber-sort | QA 3377 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 540 SK 910 SK 920 |
| ctrlnum | (OCoLC)288986457 (DE-599)BVBBV035213576 |
| dewey-full | 518.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.64 |
| dewey-search | 518.64 |
| dewey-sort | 3518.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035213576 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:28:44Z |
| institution | BVB |
| isbn | 9788847010703 9788847010710 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017019770 |
| oclc_num | 288986457 |
| open_access_boolean | |
| owner | DE-20 DE-355 DE-BY-UBR DE-703 DE-83 DE-384 DE-29T DE-188 DE-11 |
| owner_facet | DE-20 DE-355 DE-BY-UBR DE-703 DE-83 DE-384 DE-29T DE-188 DE-11 |
| physical | XVI, 601 S. Ill., graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | MS&A |
| series2 | MS&A |
| spelling | Quarteroni, Alfio 1952- Verfasser (DE-588)120370158 aut Modellistica numerica per problemi differenziali Numerical models for differential problems Alfio Quarteroni Milan [u. a.] Springer 2009 XVI, 601 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier MS&A 2 Aus d. Ital. übers. Differential equations, Partial Numerical solutions Numerical analysis Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 MS&A 2 (DE-604)BV035366143 2 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017019770&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Quarteroni, Alfio 1952- Numerical models for differential problems MS&A Differential equations, Partial Numerical solutions Numerical analysis Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4128130-5 |
| title | Numerical models for differential problems |
| title_alt | Modellistica numerica per problemi differenziali |
| title_auth | Numerical models for differential problems |
| title_exact_search | Numerical models for differential problems |
| title_full | Numerical models for differential problems Alfio Quarteroni |
| title_fullStr | Numerical models for differential problems Alfio Quarteroni |
| title_full_unstemmed | Numerical models for differential problems Alfio Quarteroni |
| title_short | Numerical models for differential problems |
| title_sort | numerical models for differential problems |
| topic | Differential equations, Partial Numerical solutions Numerical analysis Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Differential equations, Partial Numerical solutions Numerical analysis Partielle Differentialgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017019770&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035366143 |
| work_keys_str_mv | AT quarteronialfio modellisticanumericaperproblemidifferenziali AT quarteronialfio numericalmodelsfordifferentialproblems |