Dirichlet-Dirichlet domain decomposition methods for elliptic problems: h and hp finite element discretizations
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| Format: | Book |
| Language: | English |
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New Jersey
World Scientific
[2015]
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| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (pages 443-458) and index |
| Physical Description: | xx, 463 Seiten Illustrationen, Diagramme |
| ISBN: | 9789814578455 |
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MARC
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| 245 | 1 | 0 | |a Dirichlet-Dirichlet domain decomposition methods for elliptic problems |b h and hp finite element discretizations |c Vadim Glebovich Korneev, St. Petersburg State University, Russia, St. Petersburg State Polytechnical University, Russia; Ulrich Langer, Johannes Kepler University Linz, Austria, Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria |
| 264 | 1 | |a New Jersey |b World Scientific |c [2015] | |
| 264 | 4 | |c © 2015 | |
| 300 | |a xx, 463 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Includes bibliographical references (pages 443-458) and index | ||
| 650 | 4 | |a Decomposition (Mathematics) | |
| 650 | 4 | |a Differential equations, Partial | |
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| adam_text | Titel: Dirichlet-Dirichlet domain decomposition methods for elliptic problems
Autor: Korneev, Vadim G
Jahr: 2015
Contents
Preface yü
1. Introduction 1
1.1 Dirichlet-Dirichlet Domain Decomposition Methods
in Retrospect..................................................1
1.2 Two Origins of Domain Decomposition Methods..........11
2. Fundamentals of the Schwarz Methods 19
2.1 Elliptic Model Problems and their Discretizations..........19
2.2 Domain Decomposition Methods as Preconditioning ... 23
2.2.1 Simplification of Local Bilinear Forms..............23
2.2.2 Additive Domain Decomposition Algorithms ... 25
2.2.3 Inexact Subspaces Preconditioning..................29
2.2.4 Multiplicative Algorithms............................33
2.2.5 Hybrid Algorithms ..................................35
2.2.6 Jacobi and Gauss-Seidel Iterations..................36
2.3 Main Factors Influencing Convergence......................37
2.3.1 Convergence Analysis in the Case of Two
Subspaces ............................................37
2.3.2 Convergence of Additive Schwarz Algorithms ... 40
2.3.3 Exact Representation of the Minimal and Maximal
Eigenvalues of K-1K................................45
2.3.4 Convergence of Multiplicative Schwarz
Algorithms............................................46
Overlapping Domain Decomposition Methods 49
3.1 Construction Principles......................................49
3.2 Discretizations and Generalized Quasiuniformity
Conditions......................................................51
3.3 Algorithms with Generous Overlap..........................56
3.4 Loss in Convergence Due to Small Overlap ................61
3.5 Multilevel Versions............................................64
Nonoverlapping DD Methods for h FE Discretizations in 2d 71
4.1 Schur Complement Algorithms for h Discretizations ... 72
4.1.1 Exact Schur Complement Algorithms..............72
4.1.2 Schur Complement Preconditioning................75
4.2 Dirichlet-Dirichlet DD Algorithms..........................83
4.2.1 Flow Chart of DD Algorithms......................83
4.2.2 Relative Condition Number of DD
Preconditioners ......................................88
4.2.3 Discrete Low Energy Prolongations................92
4.2.4 Numerical Complexity of DD Algorithms..........98
BPS-type DD Preconditioners for 3d Elliptic Problems 101
5.1 DD Algorithms and their Main Components................105
5.1.1 Decomposition and Finite Element Meshes .... 105
5.1.2 Structure of DD Preconditioners....................108
5.1.3 Local Dirichlet Problems and Prolongation .... 111
5.1.4 Face Component......................................117
5.1.5 Wire Basket Component............................120
5.2 Condition Number and Complexity Estimates..............125
5.2.1 Abstract Bound for the Relative Condition
Number................................................125
5.2.2 Dependence of the Relative Condition Number on
Discretization and Decomposition Parameters . . 127
5.2.3 Subsidiary Results....................................130
DD Algorithms for Discretizations with Chaotically
Piecewise Variable Orthotropism 145
6.1 Single Slim Domain ..........................................148
6.1.1 Boundary Norms for Harmonic Functions..........149
6.1.2 Boundary Norms for Discrete Harmonic Functions
in Slim Rectangles.................. 159
6.1.3 Finite-Difference Shape Dependent Boundary
Norm......................... 164
6.2 Schur Complement Preconditioning by DD ........ 168
6.2.1 Preconditioning by Nonoverlapping DD
Techniques...................... 168
6.2.2 Preconditioning by Overlapping DD........ 171
6.2.3 Schur Complement Preconditioners by
Overlapping DD: Some Simplifications ...... 174
6.3 Orthotropic Discretizations with Arbitrary Aspect Ratios
on Thin Rectangles...................... 176
6.3.1 Reducing to Isotropic Discretization........ 177
6.3.2 Schur Complement Preconditioner......... 183
6.3.3 Compatible Schur Complement Preconditioner-
Solver for Subdomains ............... 185
6.4 Discretizations with Piecewise Variable Orthotropism
on Domains Composed of Shape Irregular Rectangles . . . 189
7. Nonoverlapping DD Methods for hp Discretizations of 2d
Elliptic Equations 197
7.1 Structure of DD Preconditioners and its Reflection
in the Relative Condition Number ............. 200
7.1.1 Main Components of DD Preconditioners..... 200
7.1.2 Preconditioners of Nepomnyaschikh s and
BPS Types............................................204
7.1.3 Relative Condition Number of DD
Preconditioners................... 208
7.2 Prolongations and Bounded Extension Splitting...... 218
7.2.1 A Few Remarks on Low Energy Prolongations
in Polynomial Spaces................ 218
7.2.2 Almost Discrete Harmonic Prolongations by
Means of Inexact Subdomain Dirichlet Solvers . . 221
7.3 Square Reference p-Elements, Their Stiffness and Mass
Matrices............................ 224
7.3.1 Coordinate Polynomials of Square Hierarchical
and Spectral Reference Elements.......... 227
7.3.2 Stiffness and Mass Matrices of Hierarchical
Elements....................... 230
7.4 Preconditioning of Stiffness and Mass Matrices by
Finite-Difference Matrices.................. 233
7.4.1 Preconditioners for Hierarchical Reference
Elements....................... 234
7.4.2 One-Dimensional Spectral Elements........ 239
7.4.3 Two-Dimensional Spectral Reference
Elements....................... 247
7.4.4 Factored Preconditioners for Spectral
Elements....................... 250
7.5 Schur Complement Preconditioners for Reference
Elements ........................... 255
7.5.1 Introductory Remarks................ 255
7.5.2 Polynomial Bases on the Edges .......... 256
7.5.3 Edge Schur Complement Preconditioning for
Hierarchical Coordinate Polynomials ....... 257
7.5.4 Inter-element Boundary Schur Complement
Preconditioning, Lagrange Interpolation Edge
Coordinate Functions................ 263
7.5.5 Numerical Complexity of Schur Complement
Algorithms...................... 266
7.5.6 Schur Complement Preconditioner with
Alternative Inexact Solvers............. 268
8. Fast Dirichlet Solvers for 2d Reference Elements 273
8.1 Fast Dirichlet Solvers for Hierarchical Reference
Elements ........................... 275
8.1.1 Algebraic Multigrid Solvers............. 275
8.1.2 Secondary DD Solver for the Local Dirichlet
Problems....................... 289
8.1.3 Two Main Modulus of DD Solver for Local
Dirichlet Problems.................. 307
8.1.4 Numerical Experiment with DD Solver for Local
Dirichlet Problems, Some Generalizations..... 321
8.2 Numerical Testing of DD Solver for Dirichlet Problem in a
L-Shaped Domain...................... 322
8.3 Fast Dirichlet Solvers for 2d Spectral Reference
Elements ........................... 326
8.3.1 Fast Domain Decomposition Preconditioner-
Solver......................... 326
8.3.2 Fast Multilevel Solver................................335
8.4 The Numerical Complexity of DD Methods in Two
Dimensions....................................................341
8.4.1 Hierarchical Discretizations..........................341
8.4.2 Discretizations by Spectral Elements ..............342
9. Nonoverlapping Dirichlet-Dirichlet DD Methods for hp
Discretizations of 3d Elliptic Equations 345
9.1 General Structure of DD and Schur Complement
Preconditioners........................ 346
9.1.1 Main Components of DD Preconditioners..........347
9.1.2 Abstract Bounds of Relative Spectrum of DD
Preconditioners......................................354
9.1.3 Interface Boundary Schur Complement
Preconditioning......................................356
9.1.4 Wire Basket Component............................369
9.1.5 Numerical Complexity ..............................384
9.2 Reference Elements and Finite-Difference Preconditioners 386
9.2.1 Hierarchical Reference Elements.......... 387
9.2.2 Hierarchical Elements for Adaptive
Computations.................... 392
9.2.3 Spectral Reference Elements............ 393
9.2.4 Factorized Preconditioners for Spectral Reference
Elements....................... 395
9.3 Fast Preconditioner-Solvers for Internal and
Face Problems ........................ 396
9.3.1 Multilevel Wavelet Solver of Beuchler-Schneider-
Schwab for Internal Dirichlet Problems,
Hierarchical Elements................ 397
9.3.2 Multiresolution Wavelet Solver for Faces.....401
9.3.3 Domain Decomposition Solver for Hierarchical
Reference Elements................. 403
9.3.4 Fast Domain Decomposition Solver for Spectral
Elements....................... 412
9.3.5 Multiresolution Wavelet Solver for Internal
Dirichlet Problems for Spectral Elements..... 417
Appendix A Technical Proofs 421
A.l Proof of Theorem 8.4..........................................421
Appendix B Abbreviations and Notations 437
B.l Abbreviations..................................................437
B.2 Numbers, Vectors and Matrices..............................438
B.3 Intervals, Domains, Boundaries, Derivatives, Polynomial
and Functional Spaces, and Norms..........................439
B.4 Finite Element Method ......................................440
B.5 Domain Decomposition ......................................441
Bibliography 443
Index 459
|
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| author | Korneev, Vadim G. 1937- Langer, Ulrich 1952- |
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| id | DE-604.BV042612560 |
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| language | English |
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| spelling | Korneev, Vadim G. 1937- Verfasser (DE-588)174101384 aut Dirichlet-Dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations Vadim Glebovich Korneev, St. Petersburg State University, Russia, St. Petersburg State Polytechnical University, Russia; Ulrich Langer, Johannes Kepler University Linz, Austria, Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria New Jersey World Scientific [2015] © 2015 xx, 463 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Includes bibliographical references (pages 443-458) and index Decomposition (Mathematics) Differential equations, Partial Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Dirichlet-Problem (DE-588)4129762-3 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 s Elliptische Differentialgleichung (DE-588)4014485-9 s Dirichlet-Problem (DE-588)4129762-3 s DE-604 Langer, Ulrich 1952- Verfasser (DE-588)174101392 aut HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028045411&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Korneev, Vadim G. 1937- Langer, Ulrich 1952- Dirichlet-Dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations Decomposition (Mathematics) Differential equations, Partial Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Dirichlet-Problem (DE-588)4129762-3 gnd |
| subject_GND | (DE-588)4017233-8 (DE-588)4014485-9 (DE-588)4129762-3 |
| title | Dirichlet-Dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations |
| title_auth | Dirichlet-Dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations |
| title_exact_search | Dirichlet-Dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations |
| title_full | Dirichlet-Dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations Vadim Glebovich Korneev, St. Petersburg State University, Russia, St. Petersburg State Polytechnical University, Russia; Ulrich Langer, Johannes Kepler University Linz, Austria, Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria |
| title_fullStr | Dirichlet-Dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations Vadim Glebovich Korneev, St. Petersburg State University, Russia, St. Petersburg State Polytechnical University, Russia; Ulrich Langer, Johannes Kepler University Linz, Austria, Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria |
| title_full_unstemmed | Dirichlet-Dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations Vadim Glebovich Korneev, St. Petersburg State University, Russia, St. Petersburg State Polytechnical University, Russia; Ulrich Langer, Johannes Kepler University Linz, Austria, Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy of Sciences, Linz, Austria |
| title_short | Dirichlet-Dirichlet domain decomposition methods for elliptic problems |
| title_sort | dirichlet dirichlet domain decomposition methods for elliptic problems h and hp finite element discretizations |
| title_sub | h and hp finite element discretizations |
| topic | Decomposition (Mathematics) Differential equations, Partial Finite element method Finite-Elemente-Methode (DE-588)4017233-8 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Dirichlet-Problem (DE-588)4129762-3 gnd |
| topic_facet | Decomposition (Mathematics) Differential equations, Partial Finite element method Finite-Elemente-Methode Elliptische Differentialgleichung Dirichlet-Problem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=028045411&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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