Mixed finite element methods and applications:
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| Main Authors: | , , |
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| Format: | Book |
| Language: | English |
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Berlin [u.a.]
Springer
2013
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| Series: | Springer Series in Computational Mathematics
44 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Physical Description: | XIV, 685 S. graph. Darst. 235 mm x 155 mm |
| ISBN: | 9783642365188 3642365183 |
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MARC
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| 100 | 1 | |a Boffi, Daniele |e Verfasser |0 (DE-588)135622131 |4 aut | |
| 245 | 1 | 0 | |a Mixed finite element methods and applications |c Daniele Boffi ; Franco Brezzi ; Michel Fortin |
| 264 | 1 | |a Berlin [u.a.] |b Springer |c 2013 | |
| 300 | |a XIV, 685 S. |b graph. Darst. |c 235 mm x 155 mm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Springer Series in Computational Mathematics |v 44 | |
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| 689 | 0 | 0 | |a Finite-Elemente-Methode |0 (DE-588)4017233-8 |D s |
| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Brezzi, Franco |e Verfasser |4 aut | |
| 700 | 1 | |a Fortin, Michel |e Verfasser |4 aut | |
| 830 | 0 | |a Springer Series in Computational Mathematics |v 44 |w (DE-604)BV000012004 |9 44 | |
| 856 | 4 | 2 | |m X:MVB |q text/html |u http://deposit.dnb.de/cgi-bin/dokserv?id=4240624&prov=M&dok_var=1&dok_ext=htm |3 Inhaltstext |
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| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-025977497 | |
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| _version_ | 1806324801087733760 |
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IMAGE 1
CONTENTS
1 VARIATIONAL FORMULATIONS AND FINITE ELEMENT METHODS 1
1.1 CLASSICAL METHODS 1
1.2 MODEL PROBLEMS AND ELEMENTARY PROPERTIES OF SOME FUNCTIONAL SPACES 3
1.2.1 EIGENVALUE PROBLEMS 15
1.3 DUALITY METHODS 16
1.3.1 GENERALITIES 16
1.3.2 EXAMPLES FOR SYMMETRIC PROBLEMS 19
1.3.3 DUALITY METHODS FOR NON SYMMETRIC BILINEAR FORMS . 28 1.3.4
MIXED EIGENVALUE PROBLEMS 29
1.4 DOMAIN DECOMPOSITION METHODS, HYBRID METHODS 31
1.5 MODIFIED VARIATIONAL FORMULATIONS 37
1.5.1 AUGMENTED FORMULATIONS 38
1.5.2 PERTURBED FORMULATIONS 45
1.6 BIBLIOGRAPHICAL REMARKS 46
2 FUNCTION SPACES AND FINITE ELEMENT APPROXIMATIONS 47
2.1 PROPERTIES OF THE SPACES H M (2), H( DIV; 12), AND //(CURL; Q) 47
2.1.1 BASIC PROPERTIES 47
2.1.2 PROPERTIES RELATIVE TO A PARTITION OF Q 55
2.1.3 PROPERTIES RELATIVE TO A CHANGE OF VARIABLES 58
2.1.4 DE RHAM DIAGRAM 64
2.2 FINITE ELEMENT APPROXIMATIONS OF H 1 (2) AND H 2 (Q) 65
2.2.1 CONFORMING METHODS 65
2.2.2 EXPLICIT BASIS FUNCTIONS ON TRIANGLES AND TETRAHEDRA 73 2.2.3
NONCONFORMING METHODS 74
2.2.4 QUADRILATERAL FINITE ELEMENTS ON NON AFFINE MESHES 77 2.2.5
QUADRILATERAL APPROXIMATION OF SCALAR FUNCTIONS 79
2.2.6 NON POLYNOMIAL APPROXIMATIONS 80
2.2.7 SCALING ARGUMENTS 82
VII
HTTP://D-NB.INFO/1030084386
IMAGE 2
VIII CONTENTS
2.3 SIMPLICIAL APPROXIMATIONS OF //(DIV; F2) AND //(CURL; 2) 84
2.3.1 SIMPLICIAL APPROXIMATIONS OF //(DIV; Q) 84
2.3.2 SIMPLICIAL APPROXIMATION OF //(CURL; Q) 92
2.4 APPROXIMATIONS OF H (DIV; K) ON RECTANGLES AND CUBES 96
2.4.1 RAVIART-THOMAS ELEMENTS ON RECTANGLES AND CUBES 97 2.4.2 OTHER
APPROXIMATIONS OF //(DIV; K) ON RECTANGLES 98
2.4.3 OTHER APPROXIMATIONS OF //(DIV; K) ON CUBES 101
2.4.4 APPROXIMATIONS OF //(CURL; K) ON CUBES 101
2.5 INTERPOLATION OPERATOR AND ERROR ESTIMATES 103
2.5.1 APPROXIMATIONS OF //(DIV; K) 103
2.5.2 APPROXIMATION SPACES FOR //(DIV; 2) 109
2.5.3 APPROXIMATIONS OF //(CURL; Q) 110
2.5.4 APPROXIMATION SPACES FOR //(CURL; 2) 113
2.5.5 QUADRILATERAL AND HEXAHEDRAL APPROXIMATION OF VECTOR-VALUED
FUNCTIONS IN //(DIV; Q) AND //(CURL; Z2) 114
2.5.6 DISCRETE EXACT SEQUENCES 115
2.6 EXPLICIT BASIS FUNCTIONS FOR //(DIV; K) AND //(CURL; K ) ON
TRIANGLES AND TETRAHEDRA 116
2.6.1 BASIS FUNCTIONS FOR //(DIV; K)\ THE TWO-DIMENSIONAL CASE 117
2.6.2 BASIS FUNCTIONS FOR //(DIV; K): THE THREE-DIMENSIONAL CASE 119
2.6.3 BASIS FUNCTIONS FOR //(CURL; K): THE TWO-DIMENSIONAL CASE 120
2.6.4 BASIS FUNCTIONS FOR //(CURL; K): THE THREE-DIMENSIONAL CASE 120
2.7 CONCLUDING REMARKS 121
3 ALGEBRAIC ASPECTS OF SADDLE POINT PROBLEMS 123
3.1 NOTATION, AND BASIC RESULTS IN LINEAR ALGEBRA 126
3.1.1 BASIC DEFINITIONS 126
3.1.2 SUBSPACES 127
3.1.3 ORTHOGONAL SUBSPACES 129
3.1.4 ORTHOGONAL PROJECTIONS 130
3.1.5 BASIC RESULTS 132
3.1.6 RESTRICTIONS OF OPERATORS 136
3.2 EXISTENCE AND UNIQUENESS OF SOLUTIONS: THE SOLVABILITY PROBLEM 140
3.2.1 A PRELIMINARY DISCUSSION 141
3.2.2 THE NECESSARY AND SUFFICIENT CONDITION 142
3.2.3 SUFFICIENT CONDITIONS 144
3.2.4 EXAMPLES 146
3.2.5 COMPOSITE MATRICES 148
IMAGE 3
CONTENTS IX
3.3 THE SOLVABILITY PROBLEM FOR PERTURBED MATRICES 151
3.3.1 PRELIMINARY RESULTS 151
3.3.2 MAIN RESULTS 153
3.3.3 EXAMPLES 154
3.4 STABILITY 155
3.4.1 ASSUMPTIONS ON THE NORMS 157
3.4.2 THE INF-SUP CONDITION FOR THE MATRIX B : AN ELEMENTARY DISCUSSION
161
3.4.3 THE INF-SUP CONDITION AND THE SINGULAR VALUES 164
3.4.4 THE CASE OF A ELLIPTIC ON THE WHOLE SPACE 166
3.4.5 THE CASE OF A ELLIPTIC ON THE KERNEL OF B 172
3.4.6 THE CASE OF A SATISFYING AN INF-SUP ON THE KERNEL OF B 174
3.5 ADDITIONAL RESULTS 176
3.5.1 SOME NECESSARY CONDITIONS 176
3.5.2 THE CASE OF B NOT SURJECTIVE. MODIFICATION OF THE PROBLEM 177
3.5.3 SOME SPECIAL CASES 178
3.5.4 COMPOSITE MATRICES 181
3.6 STABILITY OF PERTURBED MATRICES 183
3.6.1 THE BASIC ESTIMATE 183
3.6.2 THE SYMMETRIC CASE FOR PERTURBED MATRICES 190
4 SADDLE POINT PROBLEMS IN HILBERT SPACES 1 97
4.1 REMINDERS ON HILBERT SPACES 197
4.1.1 SCALAR PRODUCTS, NORMS, COMPLETENESS 198
4.1.2 CLOSED SUBSPACES AND DENSE SUBSPACES 201
4.1.3 ORTHOGONALITY 202
4.1.4 CONTINUOUS LINEAR OPERATORS, DUAL SPACES, POLAR SPACES 205
4.1.5 BILINEAR FORMS AND ASSOCIATED OPERATORS; TRANSPOSED OPERATORS 210
4.1.6 DUAL SPACES OF LINEAR SUBSPACES 215
4.1.7 IDENTIFICATION OF A SPACE WITH ITS DUAL SPACE 218
4.1.8 RESTRICTIONS OF OPERATORS TO CLOSED SUBSPACES 219
4.1.9 QUOTIENT SPACES 221
4.2 EXISTENCE AND UNIQUENESS OF SOLUTIONS 223
4.2.1 MIXED FORMULATIONS IN HILBERT SPACES 223
4.2.2 STABILITY CONSTANTS AND INF-SUP CONDITIONS 226
4.2.3 THE MAIN RESULT 228
4.2.4 THE CASE OF IMZ? ^ Q' 230
4.2.5 EXAMPLES 232
4.3 EXISTENCE AND UNIQUENESS FOR PERTURBED PROBLEMS 238
4.3.1 REGULAR PERTURBATIONS 238
4.3.2 SINGULAR PERTURBATIONS 257
IMAGE 4
X CONTENTS
5 APPROXIMATION OF SADDLE POINT PROBLEMS 265
5.1 BASIC RESULTS 266
5.1.1 THE BASIC ASSUMPTIONS 266
5.1.2 THE DISCRETE OPERATORS 269
5.2 ERROR ESTIMATES FOR FINITE DIMENSIONAL APPROXIMATIONS 273
5.2.1 DISCRETE STABILITY AND ERROR ESTIMATES 273
5.2.2 ADDITIONAL ERROR ESTIMATES FOR THE BASIC PROBLEM 276
5.2.3 VARIANTS OF ERROR ESTIMATES 279
5.2.4 A SIMPLE EXAMPLE 285
5.2.5 AN IMPORTANT EXAMPLE: THE PRESSURE IN THE HOMOGENEOUS STOKES
PROBLEM 293
5.3 THE CASE OF KERFL^ ^ {0} 295
5.3.1 THE CASE OF KER5,' C KER5' 295
5.3.2 THE CASE OF KERZ?^ % KERZ?' 297
5.3.3 THE CASE OF J3H OR YS/, GOING TO ZERO 299
5.4 THE INF-SUP CONDITION: CRITERIA 301
5.4.1 SOME LINGUISTIC CONSIDERATIONS 301
5.4.2 GENERAL CONSIDERATIONS 302
5.4.3 THE INF-SUP CONDITION AND THE B-COMPATIBLE INTERPOLATION OPERATOR
77/, 303
5.4.4 CONSTRUCTION OF 77/, 305
5.4.5 AN ALTERNATIVE STRATEGY: SWITCHING NORMS 306
5.5 EXTENSIONS OF ERROR ESTIMATES 309
5.5.1 PERTURBED PROBLEMS 309
5.5.2 PENALTY METHODS 312
5.5.3 SINGULAR PERTURBATIONS 315
5.5.4 NONCONFORMING METHODS 317
5.5.5 DUAL ERROR ESTIMATES 323
5.6 NUMERICAL PROPERTIES OF THE DISCRETE PROBLEM 326
5.6.1 THE MATRIX FORM OF THE DISCRETE PROBLEM 327
5.6.2 AND IF THE INF-SUP CONDITION DOES NOT HOLD? 329
5.6.3 SOLUTION METHODS 331
5.7 CONCLUDING REMARKS 335
6 COMPLEMENTS: STABILISATION METHODS, EIGENVALUE PROBLEMS 337 6.1
AUGMENTED FORMULATIONS 337
6.1.1 AN ABSTRACT FRAMEWORK FOR STABILISED METHODS 337
6.1.2 STABILISING TERMS 339
6.1.3 STABILITY CONDITIONS FOR AUGMENTED FORMULATIONS 342 6.1.4
DISCRETISATIONS OF AUGMENTED FORMULATIONS 346
6.1.5 STABILISING WITH THE "ELEMENT-WISE EQUATIONS" 350
6.2 OTHER STABILISATIONS 355
6.2.1 GENERAL STABILITY CONDITIONS 355
6.2.2 STABILITY OF DISCRETISED FORMULATIONS 358
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CONTENTS XI
6.3 MINIMAL STABILISATIONS 360
6.3.1 ANOTHER FORM OF MINIMAL STABILISATION 374
6.4 ENHANCED STRAIN METHODS 379
6.5 EIGENVALUE PROBLEMS 381
6.5.1 SOME CLASSICAL RESULTS 384
6.5.2 EIGENVALUE PROBLEMS IN MIXED FORM 385
6.5.3 SPECIAL RESULTS FOR PROBLEMS OF TYPE ( F 0) AND(0,G) 387
6.5.4 EIGENVALUE PROBLEMS OF THE TYPE (/, 0) 389
6.5.5 EIGENVALUE PROBLEMS OF THE FORM (0, G) 392
7 MIXED METHODS FOR ELLIPTIC PROBLEMS 401
7.1 NON-STANDARD METHODS FOR DIRICHLET'S PROBLEM 401
7.1.1 DESCRIPTION OF THE PROBLEM 401
7.1.2 MIXED FINITE ELEMENT METHODS FOR DIRICHLET'S PROBLEM 403
7.1.3 EIGENVALUE PROBLEM FOR THE MIXED FORMULATION 408
7.1.4 PRIMAL HYBRID METHODS 410
7.1.5 PRIMAL MACRO-HYBRID METHODS AND DOMAIN DECOMPOSITIONS 419
7.1.6 DUAL HYBRID METHODS 420
7.2 NUMERICAL SOLUTIONS 426
7.2.1 PRELIMINARIES 426
7.2.2 INTER-ELEMENT MULTIPLIERS 426
7.3 A BRIEF ANALYSIS OF THE COMPUTATIONAL EFFORT 430
7.4 ERROR ANALYSIS FOR THE MULTIPLIER 432
7.5 ERROR ESTIMATES IN OTHER NORMS 437
7.6 APPLICATION TO AN EQUATION ARISING FROM SEMICONDUCTOR THEORY 439
7.7 USING ANISOTROPIC MESHES 441
7.8 RELATIONS WITH FINITE VOLUME METHODS 445
7.8.1 THE ONE AND TWO-DIMENSIONAL CASES 446
7.8.2 THE TWO-DIMENSIONAL CASE 447
7.8.3 THE THREE-DIMENSIONAL CASE 452
7.9 NONCONFORMING METHODS: A TRAP TO AVOID 453
7.10 AUGMENTED FORMULATIONS (GALERKIN LEAST SQUARES METHODS) 455 7.11 A
POSTERIORI ERROR ESTIMATES 457
8 INCOMPRESSIBLE MATERIALS AND FLOW PROBLEMS 459
8.1 INTRODUCTION 460
8.2 THE STOKES PROBLEM AS A MIXED PROBLEM 462
8.2.1 MIXED FORMULATION 462
8.3 SOME EXAMPLES OF FAILURE AND EMPIRICAL CURES 466
8.3.1 CONTINUOUS PRESSURE: THE .P, - P\ ELEMENT 466
8.3.2 DISCONTINUOUS PRESSURE: THE P_ { - PO APPROXIMATION 467
IMAGE 6
XII CONTENTS
8.4 BUILDING A B-COMPATIBLE OPERATOR: THE SIMPLEST
STABLE ELEMENTS 468
8.4.1 BUILDING A B-COMPATIBLE OPERATOR 469
8.4.2 A STABLE CASE: THE MINI ELEMENT 470
8.4.3 ANOTHER STABLE APPROXIMATION: THE BI-DIMENSIONAL P_ 2 ~ ? O
ELEMENT 471
8.4.4 THE NONCONFORMING P_\ - P Q APPROXIMATION 475
8.5 OTHER TECHNIQUES FOR CHECKING THE INF-SUP CONDITION 477
8.5.1 PROJECTION ONTO CONSTANTS 477
8.5.2 VERFURTH'S TRICK 478
8.5.3 SPACE AND DOMAIN DECOMPOSITION TECHNIQUES 480
8.5.4 MACRO-ELEMENT TECHNIQUE 482
8.5.5 MAKING USE OF THE INTERNAL DEGREES OF FREEDOM 484
8.6 TWO-DIMENSIONAL STABLE ELEMENTS 486
8.6.1 CONTINUOUS PRESSURE ELEMENTS 487
8.6.2 DISCONTINUOUS PRESSURE ELEMENTS 488
8.6.3 QUADRILATERAL ELEMENTS, Q H - PK-\ ELEMENTS 489
8.7 THREE-DIMENSIONAL STABLE ELEMENTS 491
8.7.1 CONTINUOUS PRESSURE 3-D ELEMENTS 491
8.7.2 DISCONTINUOUS PRESSURE 3-D ELEMENTS 491
8.8 PJ - PK~\ SCHEMES AND GENERALISED HOOD-TAYLOR ELEMENTS 494 8.8.1
DISCONTINUOUS PRESSURE PJ. - PK-\ ELEMENTS 494
8.8.2 GENERALISED HOOD-TAYLOR ELEMENTS 496
8.9 OTHER DEVELOPMENTS FOR DIVERGENCE-FREE STOKES APPROXIMATION AND MASS
CONSERVATION 504
8.9.1 EXACTLY DIVERGENCE-FREE STOKES ELEMENTS, DISCONTINUOUS GALERKIN
METHODS 505
8.9.2 STOKES ELEMENTS ALLOWING FOR ELEMENT-WISE MASS CONSERVATION 506
8.10 SPURIOUS PRESSURE MODES 507
8.10.1 LIVING WITH SPURIOUS PRESSURE MODES: PARTIAL CONVERGENCE 510
8.10.2 THE BILINEAR VELOCITY-CONSTANT PRESSURE Q | - P Q ELEMENT 511
8.11 EIGENVALUE PROBLEMS 517
8.12 NEARLY INCOMPRESSIBLE ELASTICITY, REDUCED INTEGRATION METHODS AND
RELATION WITH PENALTY METHODS 519
8.12.1 VARIATIONAL FORMULATIONS AND ADMISSIBLE DISCRETISATIONS 519
8.12.2 REDUCED INTEGRATION METHODS 520
8.12.3 EFFECTS OF INEXACT INTEGRATION 523
8.13 OTHER STABILISATION PROCEDURES 527
8.13.1 AUGMENTED METHOD FOR THE STOKES PROBLEM 528
8.13.2 DEFINING AN APPROXIMATE INVERSE 530
8.13.3 MINIMAL STABILISATIONS FOR STOKES 534
IMAGE 7
CONTENTS XIII
8.14 CONCLUDING REMARKS: CHOICE OF ELEMENTS 537
8.14.1 CHOICE OF ELEMENTS 537
9 COMPLEMENTS ON ELASTICITY PROBLEMS 539
9.1 INTRODUCTION 539
9.1.1 CONTINUOUS FORMULATION OF STRESS METHODS 540
9.1.2 NUMERICAL APPROXIMATIONS OF STRESS FORMULATIONS 543 9.2 RELAXED
SYMMETRY 544
9.3 TENSORS, TENSORIAL NOTATION AND RESULTS ON SYMMETRY 544
9.3.1 CONTINUOUS FORMULATION OF THE RELAXED SYMMETRY APPROACH 548
9.3.2 NUMERICAL APPROXIMATION OF RELAXED-SYMMETRY FORMULATIONS 551
9.4 SOME FAMILIES OF METHODS WITH REDUCED SYMMETRY 555
9.4.1 METHODS BASED ON STOKES ELEMENTS 555
9.4.2 STABILISATION BY //(CURL) BUBBLES 558
9.4.3 TWO EXAMPLES 561
9.4.4 METHODS BASED ON THE PROPERTIES OF 17^ 563
9.5 LOOSING THE INCLUSION OF KERNEL: STABILISED METHODS 567
9.6 CONCLUDING REMARKS 572
10 COMPLEMENTS ON PLATE PROBLEMS 575
10.1 A MIXED FOURTH-ORDER PROBLEM 575
10.1.1 THE IFR - CO BIHARMONIC PROBLEM 575
10.1.2 EIGENVALUES OF THE BIHARMONIC PROBLEM 578
10.2 DUAL HYBRID METHODS FOR PLATE BENDING PROBLEMS 579
10.3 MIXED METHODS FOR LINEAR THIN PLATES 588
10.4 MODERATELY THICK PLATES 596
10.4.1 GENERALITIES 596
10.4.2 THE MATHEMATICAL FORMULATION 598
10.4.3 MIXED FORMULATION OF THE MINDLIN-REISSNER MODEL 600 10.4.4 A
DECOMPOSITION PRINCIPLE AND THE STOKES CONNECTION 606
10.4.5 DISCRETISATION OF THE PROBLEM 609
10.4.6 CONTINUOUS PRESSURE APPROXIMATIONS 622
10.4.7 DISCONTINUOUS PRESSURE ELEMENTS 622
11 MIXED FINITE ELEMENTS FOR ELECTROMAGNETIC PROBLEMS 625
11.1 USEFUL RESULTS ABOUT THE SPACE //(CURL; ?), ITS BOUNDARY TRACES,
AND THE DE RHAM COMPLEX 626
11.1.1 THE DE RHAM COMPLEX AND THE HELMHOLTZ DECOMPOSITION WHEN Q IS
SIMPLY CONNECTED 626
11.1.2 THE FRIEDRICHS INEQUALITY 627
11.1.3 EXTENSION TO MORE GENERAL TOPOLOGIES 627
11.1.4 //(CURL; Q) IN TWO SPACE DIMENSIONS 628
IMAGE 8
XIV CONTENTS
11.2 THE TIME HARMONIC MAXWELL SYSTEM 629
11.2.1 MAXWELL'S EIGENVALUE PROBLEM 630
11.2.2 ANALYSIS OF THE TIME HARMONIC MAXWELL SYSTEM 633
11.2.3 APPROXIMATION OF THE TIME HARMONIC MAXWELL EQUATIONS 636
11.3 APPROXIMATION OF THE MAXWELL EIGENVALUE PROBLEM 639
11.3.1 ANALYSIS OF THE TWO-DIMENSIONAL CASE 641
11.3.2 DISCRETE COMPACTNESS PROPERTY 644
11.3.3 NODAL FINITE ELEMENTS 647
11.3.4 EDGE FINITE ELEMENTS 653
11.4 ENFORCING THE DIVERGENCE-FREE CONDITION BY A PENALTY METHOD 654
11.5 SOME REMARKS ON EXTERIOR CALCULUS 658
11.6 CONCLUDING REMARKS 662
REFERENCES 663
INDEX 681 |
| any_adam_object | 1 |
| author | Boffi, Daniele Brezzi, Franco Fortin, Michel |
| author_GND | (DE-588)135622131 |
| author_facet | Boffi, Daniele Brezzi, Franco Fortin, Michel |
| author_role | aut aut aut |
| author_sort | Boffi, Daniele |
| author_variant | d b db f b fb m f mf |
| building | Verbundindex |
| bvnumber | BV040999876 |
| classification_rvk | SI 990 SK 910 SK 920 |
| ctrlnum | (OCoLC)855555724 (DE-599)DNB1030084386 |
| dewey-full | 518.25 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518.25 |
| dewey-search | 518.25 |
| dewey-sort | 3518.25 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV040999876 |
| illustrated | Illustrated |
| indexdate | 2024-08-03T00:39:28Z |
| institution | BVB |
| isbn | 9783642365188 3642365183 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-025977497 |
| oclc_num | 855555724 |
| open_access_boolean | |
| owner | DE-188 DE-11 DE-706 DE-29T DE-355 DE-BY-UBR DE-83 DE-20 |
| owner_facet | DE-188 DE-11 DE-706 DE-29T DE-355 DE-BY-UBR DE-83 DE-20 |
| physical | XIV, 685 S. graph. Darst. 235 mm x 155 mm |
| publishDate | 2013 |
| publishDateSearch | 2013 |
| publishDateSort | 2013 |
| publisher | Springer |
| record_format | marc |
| series | Springer Series in Computational Mathematics |
| series2 | Springer Series in Computational Mathematics |
| spelling | Boffi, Daniele Verfasser (DE-588)135622131 aut Mixed finite element methods and applications Daniele Boffi ; Franco Brezzi ; Michel Fortin Berlin [u.a.] Springer 2013 XIV, 685 S. graph. Darst. 235 mm x 155 mm txt rdacontent n rdamedia nc rdacarrier Springer Series in Computational Mathematics 44 Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 s DE-604 Brezzi, Franco Verfasser aut Fortin, Michel Verfasser aut Springer Series in Computational Mathematics 44 (DE-604)BV000012004 44 X:MVB text/html http://deposit.dnb.de/cgi-bin/dokserv?id=4240624&prov=M&dok_var=1&dok_ext=htm Inhaltstext DNB Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025977497&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Boffi, Daniele Brezzi, Franco Fortin, Michel Mixed finite element methods and applications Springer Series in Computational Mathematics Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| subject_GND | (DE-588)4017233-8 |
| title | Mixed finite element methods and applications |
| title_auth | Mixed finite element methods and applications |
| title_exact_search | Mixed finite element methods and applications |
| title_full | Mixed finite element methods and applications Daniele Boffi ; Franco Brezzi ; Michel Fortin |
| title_fullStr | Mixed finite element methods and applications Daniele Boffi ; Franco Brezzi ; Michel Fortin |
| title_full_unstemmed | Mixed finite element methods and applications Daniele Boffi ; Franco Brezzi ; Michel Fortin |
| title_short | Mixed finite element methods and applications |
| title_sort | mixed finite element methods and applications |
| topic | Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| topic_facet | Finite-Elemente-Methode |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=4240624&prov=M&dok_var=1&dok_ext=htm http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025977497&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000012004 |
| work_keys_str_mv | AT boffidaniele mixedfiniteelementmethodsandapplications AT brezzifranco mixedfiniteelementmethodsandapplications AT fortinmichel mixedfiniteelementmethodsandapplications |