Multilevel Block Factorization Preconditioners: Matrix-based Analysis and Algorithms for Solving Finite Element Equations
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
New York, NY
Springer New York
2008
|
| Schlagworte: | |
| Online-Zugang: | http://digitool.hbz-nrw.de:1801/webclient/DeliveryManager?pid=2471487&custom_att_2=simple_viewer Inhaltsverzeichnis |
| Beschreibung: | XIV, 529 S. Ill., graph. Darst. 235 mm x 155 mm |
| ISBN: | 9780387715636 0387715630 |
Internformat
MARC
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| 016 | 7 | |a 985875062 |2 DE-101 | |
| 020 | |a 9780387715636 |c Gb. : ca. EUR 82.39 (freier Pr.), ca. sfr 134.00 (freier Pr.) |9 978-0-387-71563-6 | ||
| 020 | |a 0387715630 |c Gb. : ca. EUR 82.39 (freier Pr.), ca. sfr 134.00 (freier Pr.) |9 0-387-71563-0 | ||
| 024 | 3 | |a 9780387715636 | |
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| 100 | 1 | |a Vasilevski, Panayot |e Verfasser |0 (DE-588)12078372X |4 aut | |
| 245 | 1 | 0 | |a Multilevel Block Factorization Preconditioners |b Matrix-based Analysis and Algorithms for Solving Finite Element Equations |c Panayot Vassilevski |
| 264 | 1 | |a New York, NY |b Springer New York |c 2008 | |
| 300 | |a XIV, 529 S. |b Ill., graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 650 | 4 | |a Ecuaciones diferenciales lineales - Soluciones numericas | |
| 650 | 4 | |a Ecuaciones diferenciales parciales - Soluciones numéricas | |
| 650 | 4 | |a Método de elementos finitos | |
| 650 | 0 | 7 | |a Mehrgitterverfahren |0 (DE-588)4038376-3 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-017135639 | ||
Datensatz im Suchindex
| _version_ | 1804138643580780544 |
|---|---|
| adam_text | Contents
Preface
............................................................
vii
Part I Motivation for Preconditioning
1
A Finite Element Tutorial
....................................... 3
1.1
Finite element matrices
...................................... 3
1.2
Finite element refinement
.................................... 9
1.3
Coarse-grid approximation
................................... 10
1.4
The mass (Gram) matrix
..................................... 15
1.5
A strong approximation property
............................ 18
1.6
The coarse-grid correction
................................... 21
1.7
A f.e. (geometric) two-grid method
............................ 22
1.8
Element matrices and matrix
orderings
......................... 25
1.9
Element topology
.......................................... 29
1.9.1
Main definitions and constructions
...................... 30
1.9.2
Element faces
....................................... 32
1.9.3
Faces of AEs
........................................ 34
1.9.4
Edges of AEs
....................................... 35
1.9.5
Vertices of AEs
...................................... 36
1.9.6
Nested dissection ordering
............................ 36
1.9.7
Element agglomeration algorithms
...................... 37
1.10
Finite element matrices on many processors
.................... 39
1.11
The mortar method
......................................... 41
1.11.1
Algebraic construction of mortar spaces
................. 43
2
A Main Goal
.................................................. 49
Contents
Part II
Block
Factorization Preconditioners
3 Two-by-Two
Block Matrices and Their Factorization
............... 55
3.1
Matrices of two-by-two block form
............................ 55
3.1.1
Exact block-factorization.
Schur
complements
............ 55
3.1.2
Kato s Lemma
....................................... 60
3.1.3
Convergent iteration in A-norm
........................ 61
3.2
Approximate block-factorization
.............................. 63
3.2.1
Product iteration matrix formula
........................ 63
3.2.2
Block-factorizations and product iteration methods
........ 65
3.2.3
Definitions of two-level Btl and two-grid Bjg
preconditioners
...................................... 67
3.2.4
A main identity
...................................... 68
3.2.5
A simple lower-bound estimate
........................ 70
3.2.6
Sharp upper bound
................................... 70
3.2.7
The sharp spectral equivalence result
.................... 72
3.2.8
Analysis of BTL
..................................... 74
3.2.9
Analysis of BTG
..................................... 75
3.3
Algebraic two-grid methods and preconditioners;
sufficient conditions for spectral equivalence
.................... 78
3.4
Classical two-level block-factorization preconditioners
........... 81
3.4.1
A general procedure of generating stable block-matrix
partitioning
......................................... 84
4
Classical Examples of Block-Factorizations
....................... 89
4.1
Block-ILU factorizations
.................................... 89
4.2
The M-matrix case
......................................... 92
4.3
Decay rates of inverses of band matrices
....................... 98
4.4
Algorithms for approximate band inverses
...................... 103
4.5
Wittum s frequency filtering decomposition
.................... 109
4.6
Block-ILU factorizations with block-size reduction
.............. 113
4.7
An alternative approximate block-LU factorization
.............. 117
4.8
Odd-even modified block-ILU methods
........................ 122
4.9
A nested dissection (approximate) inverse
...................... 125
5
Multigrid (MG)
................................................ 129
5.1
From two-grid to multigrid
................................... 129
5.2
MG as block Gauss-Seidel
................................... 133
5.3
A MG analysis in general terms
............................... 134
5.4
The XZ identity
............................................ 140
5.5
Some classical upper bounds
................................. 144
5.5.1
Variable V-cycle
..................................... 152
5.6
MG with more recursive cycles; W-cycle
....................... 157
5.6.1
Definition of a
υ
-fold
MG-cycle; complexity
............. 157
5.6.2
AMLI-cycle multigrid
................................ 158
Contents xi
5.6.3
Analysis of AMLI
................................... 159
5.6.4
Complexity of the AMLI-cycle
......................... 162
5.6.5
Optimal W-cycle methods
............................. 163
5.7
MG and additive MG
....................................... 165
5.7.1
The BPX-preconditioner
.............................. 165
5.7.2
Additive representation of MG
......................... 166
5.7.3
Additive MG; convergence properties
................... 167
5.7.4
MG convergence based on results for matrix subblocks
___ 174
5.8
Cascadic multigrid
......................................... 177
5.8.1
Convergence in a stronger norm
........................ 182
5.9
The hierarchical basis (HB) method
........................... 185
5.9.1
The additive multilevel HB
............................ 185
5.9.2
A stable multilevel hierarchical (direct) decomposition
..... 188
5.9.3
Approximation of Z^-projections
....................... 192
5.9.4
Construction of bases in the coordinate spaces
............ 195
5.9.5
The approximate wavelet hierarchical basis (or AWHB)
___ 196
Topics on Algebraic Multigrid (AMG)
............................ 199
6.1
Motivation for the construction of
Ρ
........................... 199
6.2
On the classical AMG construction of
Ρ
....................... 202
6.3
On the constrained trace minimization construction of
Ρ
.......... 205
6.4
On the coarse-grid selection
.................................. 207
6.5
On the sparsity pattern of
Ρ
.................................. 207
6.6
Coarsening by compatible relaxation
.......................... 208
6.6.1
Smoothing property and compatible relaxation
............ 209
6.6.2
Using inexact projections
............................. 211
6.7
The need for adaptive AMG
.................................. 213
6.8
Smoothing based on V- T relaxation
........................ 214
6.9
AMGe: An element agglomeration AMG
...................... 225
6.9.1
Element-based construction of
Ρ
....................... 226
6.9.2
On various norm bounds of
Ρ
.......................... 228
6.10
Multivector fitting interpolation
............................... 234
6.11
Window-based spectral AMG
................................ 235
6.12
Two-grid convergence of vector-preserving AMG
............... 241
6.13
The result of
Vaněk,
Brezina,
and
Mandel......................
249
6.13.1
Null vector-based polynomially smoothed bases
.......... 249
6.13.2
Some properties of Chebyshev-like polynomials
.......... 252
6.13.3
A general setting forthe
SA
method
.................... 255
Domain Decomposition (DD) Methods
............................ 263
7.1
Nonoverlapping blocks
...................................... 263
7.2
Boundary extension mappings based on solving special coarse
problems
.................................................. 264
7.3
Weakly overlapping blocks
.................................. 267
7.4
Classical domain-embedding
(DE) preconditioners
.............. 270
xii Contents
7.5 DE preconditioners
without extension mappings
................. 272
7.6 Fast
solvers for tensor product matrices
........................ 274
7.7 Schwarz
methods
........................................... 280
7.8
Additive
Schwarz
preconditioners
............................. 286
7.9
The domain decomposition paradigm of Bank and Hoist
.......... 291
7.9.1
Local error estimates
................................. 300
7.10
The
FAC
method and related preconditioning
................... 304
7.11
Auxiliary space preconditioning methods
....................... 313
8
Preconditioning Nonsymmetric and Indefinite Matrices
............ 319
8.1
An abstract setting
.......................................... 319
8.2
A perturbation point of view
................................. 323
8.3
Implementation
............................................ 325
9
Preconditioning Saddle-Point Matrices
........................... 327
9.1
Basic properties of saddle-point matrices
....................... 327
9.2
S.p.d. preconditioners
....................................... 330
9.2.1
Preconditioning based on inf-sup condition
............ 332
9.3
Transforming
Λ
to a positive definite matrix
.................... 339
9.4
(Inexact) Uzawa and distributive relaxation methods
............. 341
9.4.1
Distributive relaxation
................................ 341
9.4.2
The Bramble-Pasciak transformation
................... 342
9.4.3
A note on two-grid analysis
............................ 344
9.4.4
Inexact Uzawa methods
............................... 347
9.5
A constrained minimization approach
.......................... 353
10
Variable-Step Iterative Methods
................................. 363
10.1
Variable-step (nonlinear) preconditioners
....................... 363
10.2
Variable-step preconditioned
CG
method
....................... 365
10.3
Variable-step multilevel preconditioners
........................ 371
10.4
Variable-step AMLI-cycle MG
............................... 372
11
Preconditioning Nonlinear Problems
............................. 377
11.1
Problem formulation
........................................ 377
1
1.2
Choosing an accurate initial approximation
..................... 379
11.3
The inexact Newton algorithm
................................ 380
12
Quadratic Constrained Minimization Problems
................... 385
12.1
Problem formulation
........................................ 385
12.1.1
Projection methods
................................... 386
12.1.2
A modified projection method
......................... 389
12.2
Computable projections
..................................... 390
12.3
Dual problem approach
...................................... 391
12.3.1
Dual problem formulation
............................. 391
12.3.2
Reduced problem formulation
.......................... 393
Contents xiii
12.4
A
monotone
two-grid scheme
................................ 397
12.4.1
Projected Gauss-Seidel
............................... 398
12.4.2
Coarse-grid solution
.................................. 398
12.5
A monotone FAS constrained minimization algorithm
............ 401
Partili
Appendices
A Generalized Conjugate Gradient Methods
........................ 407
A.
1
A general variational setting for solving nonsymmetric problems
... 407
A.2 A quick
CG
guide
.......................................... 410
A.2.1 The
CG
algorithm
.................................... 410
A.2.2 Preconditioning
...................................... 411
A.2.3 Best polynomial approximation property of
CG........... 412
A.2.4 A decay rate estimate for A~l
.......................... 412
В
Properties of Finite Element Matrices. Further Details
............. 415
B.
1
Piecewise linear finite elements
............................... 415
B.2
A
semilinear
second-order elliptic PDE
........................ 429
B.3 Stable two-level HB decomposition of finite element spaces
....... 433
B.3.1 A two-level hierarchical basis and related strengthened
Cauchy-Schwarz inequality
........................... 433
B.3.2 On the MG convergence uniform w.r.t. the mesh and
jumps in the PDE coefficients
.......................... 438
B.4 Mixed methods for second-order elliptic PDEs
.................. 439
B.5 Nonconforming elements and Stokes problem
................... 448
B.6 F.e. discretization of Maxwell s equations
...................... 453
С
Computable Scales of Sobolev Norms
............................. 457
С
1
tf^stable decompositions
................................... 457
C.2 Preliminary facts
........................................... 457
C.3 The main norm equivalence result
............................. 459
C.4 The uniform coercivity property
.............................. 463
D
Multilevel Algorithms for Boundary Extension Mappings
........... 467
Е Яд1
-norm Characterization
...................................... 471
E.I Optimality of the L2-proJections
..............................
471
E.
1.1
Яд
-stable decompositions of finite element functions
...... 475
F
MG
Convergence Results for Finite Element Problems
............. 477
F.
1
Requirements on the multilevel f.e. decompositions for the MG
convergence analysis
........................................ 479
F.2 A MG for weighted tf
(curi)
space
............................ 487
F.2.
1
A multilevel decomposition of weighted
Nédélec
spaces
... 489
F.3 A multilevel decomposition of div-free Raviart-Thomas spaces
---- 495
F.4 A multilevel decomposition of weighted tfidivj-space
........... 499
xiv Contents
G
Some Auxiliary Inequalities
..................................... 507
G.
1
Kantorovich s inequality
..................................... 507
G.2 An inequality between powers of matrices
...................... 508
G.3 Energy bound of the nodal interpolation operator
................ 510
References
......................................................... 513
Index
.............................................................. 527
|
| any_adam_object | 1 |
| author | Vasilevski, Panayot |
| author_GND | (DE-588)12078372X |
| author_facet | Vasilevski, Panayot |
| author_role | aut |
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| classification_rvk | SK 920 |
| ctrlnum | (OCoLC)427558562 (DE-599)DNB985875062 |
| dewey-full | 530.155353 515.354 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics 515 - Analysis |
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| dewey-search | 530.155353 515.354 |
| dewey-sort | 3530.155353 |
| dewey-tens | 530 - Physics 510 - Mathematics |
| discipline | Physik Mathematik Medizin |
| format | Book |
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| id | DE-604.BV035331208 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:31:27Z |
| institution | BVB |
| isbn | 9780387715636 0387715630 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017135639 |
| oclc_num | 427558562 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-706 DE-83 DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-706 DE-83 DE-11 |
| physical | XIV, 529 S. Ill., graph. Darst. 235 mm x 155 mm |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Springer New York |
| record_format | marc |
| spelling | Vasilevski, Panayot Verfasser (DE-588)12078372X aut Multilevel Block Factorization Preconditioners Matrix-based Analysis and Algorithms for Solving Finite Element Equations Panayot Vassilevski New York, NY Springer New York 2008 XIV, 529 S. Ill., graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Ecuaciones diferenciales lineales - Soluciones numericas Ecuaciones diferenciales parciales - Soluciones numéricas Método de elementos finitos Mehrgitterverfahren (DE-588)4038376-3 gnd rswk-swf Gebietszerlegungsmethode (DE-588)4309232-9 gnd rswk-swf Mehrgitterverfahren (DE-588)4038376-3 s Gebietszerlegungsmethode (DE-588)4309232-9 s DE-604 http://digitool.hbz-nrw.de:1801/webclient/DeliveryManager?pid=2471487&custom_att_2=simple_viewer Verlagsdaten Springer Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017135639&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Vasilevski, Panayot Multilevel Block Factorization Preconditioners Matrix-based Analysis and Algorithms for Solving Finite Element Equations Ecuaciones diferenciales lineales - Soluciones numericas Ecuaciones diferenciales parciales - Soluciones numéricas Método de elementos finitos Mehrgitterverfahren (DE-588)4038376-3 gnd Gebietszerlegungsmethode (DE-588)4309232-9 gnd |
| subject_GND | (DE-588)4038376-3 (DE-588)4309232-9 |
| title | Multilevel Block Factorization Preconditioners Matrix-based Analysis and Algorithms for Solving Finite Element Equations |
| title_auth | Multilevel Block Factorization Preconditioners Matrix-based Analysis and Algorithms for Solving Finite Element Equations |
| title_exact_search | Multilevel Block Factorization Preconditioners Matrix-based Analysis and Algorithms for Solving Finite Element Equations |
| title_full | Multilevel Block Factorization Preconditioners Matrix-based Analysis and Algorithms for Solving Finite Element Equations Panayot Vassilevski |
| title_fullStr | Multilevel Block Factorization Preconditioners Matrix-based Analysis and Algorithms for Solving Finite Element Equations Panayot Vassilevski |
| title_full_unstemmed | Multilevel Block Factorization Preconditioners Matrix-based Analysis and Algorithms for Solving Finite Element Equations Panayot Vassilevski |
| title_short | Multilevel Block Factorization Preconditioners |
| title_sort | multilevel block factorization preconditioners matrix based analysis and algorithms for solving finite element equations |
| title_sub | Matrix-based Analysis and Algorithms for Solving Finite Element Equations |
| topic | Ecuaciones diferenciales lineales - Soluciones numericas Ecuaciones diferenciales parciales - Soluciones numéricas Método de elementos finitos Mehrgitterverfahren (DE-588)4038376-3 gnd Gebietszerlegungsmethode (DE-588)4309232-9 gnd |
| topic_facet | Ecuaciones diferenciales lineales - Soluciones numericas Ecuaciones diferenciales parciales - Soluciones numéricas Método de elementos finitos Mehrgitterverfahren Gebietszerlegungsmethode |
| url | http://digitool.hbz-nrw.de:1801/webclient/DeliveryManager?pid=2471487&custom_att_2=simple_viewer http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017135639&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT vasilevskipanayot multilevelblockfactorizationpreconditionersmatrixbasedanalysisandalgorithmsforsolvingfiniteelementequations |