Least-squares finite element methods:
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| Language: | English |
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New York, NY
Springer
2009
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| Series: | Applied mathematical sciences
166 |
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| Physical Description: | XXII, 660 S. graph. Darst. |
| ISBN: | 9780387308883 |
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| 100 | 1 | |a Bochev, Pavel B. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Least-squares finite element methods |c Pavel B. Bochev ; Max D. Gunzburger |
| 246 | 1 | 3 | |a Least squares finite element methods |
| 264 | 1 | |a New York, NY |b Springer |c 2009 | |
| 300 | |a XXII, 660 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applied mathematical sciences |v 166 | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 4 | |a Finite element method | |
| 650 | 4 | |a Least squares | |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Gunzburger, Max D. |e Verfasser |4 aut | |
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Contents
Part I Survey of Variations! Principles and Associated Finite Element
Methods
1
Classical Variational Methods
.
1.1
1.2
1.3
1.4
. 3
Variational Methods for Operator Equations
. 4
A Taxonomy of Classical Variational Formulations
. 8
1.2.1
Weakly Coercive Problems
. 8
1.2.2
Strongly Coercive Problems
. 9
1.2.3
Mixed Variational Problems
. 10
1.2.4
Relations Between Variational Problems
and Optimization Problems
. 12
Approximation of Solutions of Variational Problems
. 15
.3.1
Weakly and Strongly Coercive Variational Problems
. 15
.3.2
Mixed Variational Problems
. 18
Examples
. 22
.4.1
The
Poisson
Equation
. 22
.4.2
The Equations of Linear Elasticity
. 25
.4.3
The Stokes Equations
. 26
.4.4
The Helmholtz Equation
. 28
.4.5
A Scalar Linear Advection-Diffusion-Reaction Equation
. 30
.4.6
The Navier-Stokes Equations
. 30
1.5
A Comparative Summary of Classical Finite Element Methods
. 31
2
Alternative Variational Formulations
. 35
2.1
Modified Variational Principles
. 36
2.1.1
Enhanced and Stabilized Methods for Weakly Coercive
Problems
. 36
2.1.2
Stabilized Methods for Strongly Coercive Problems
. 46
2.2
Least-Squares Principles
. 49
2.2.1
A Straightforward Least-Squares Finite Element Method
. 51
2.2.2
Practical Least-Squares Finite Element Methods
. 53
xvi Contents
2.2.3
Norm-Equivalence
Versus
Practicality
. 58
2.2.4
Some Questions and Answers
. 60
2.3
Putting Things in Perspective and What to Expect from the Book
. 62
Part II Abstract Theory of Least-Squares Finite Element Methods
3
Mathematical Foundations of Least-Squares Finite Element Methods
69
3.1
Least-Squares Principles for Linear Operator Equations
in Hubert Spaces
. 70
3.1.1
Problems with Zero Nullity
. 71
3.1.2
Problems with Positive Nullity
. 73
3.2
Application to Partial Differential Equations
. 75
3.2.1
Energy Balances
. 76
3.2.2
Continuous Least-Squares Principles
. 77
3.3
General Discrete Least-Squares Principles
. 80
3.3.1
Error Analysis
. 82
3.3.2
The Need for Continuous Least-Squares Principles
. 84
3.4
Binding Discrete Least-Squares Principles to Partial Differential
Equations
. 85
3.4.1
Transformations from Continuous to Discrete
Least-Squares Principles
. 86
3.5
Taxonomy of Conforming Discrete Least-Squares Principles
and their Analysis
. 90
3.5.1
Compliant Discrete Least-Squares Principles
. 92
3.5.2
Norm-Equivalent Discrete Least-Squares Principles
. 94
3.5.3
Quasi-Norm-Equivalent Discrete Least-Squares Principles
96
3.5.4
Summary Review of Discrete Least-Squares Principles
.100
4
The Agmon-Douglis-Nirenberg Setting for Least-Squares Finite
Element Methods
.103
4.1
Transformations to First-Order Systems
.105
4.2
Energy Balances
.106
4.2.1
Homogeneous Elliptic Systems
.107
4.2.2
Non-Homogeneous Elliptic Systems
.107
4.3
Continuous Least-Squares Principles
.108
4.3.1
Homogeneous Elliptic Systems
.108
4.3.2
Non-Homogeneous Elliptic Systems
.110
4.4
Least-Squares Finite Element Methods for Homogeneous
Elliptic Systems
.112
4.5
Least-Squares Finite Element Methods for Non-Homogeneous
Elliptic Systems
.114
4.5.1
Quasi-Norm-Equivalent Discrete Least-Squares Principles
114
4.5.2
Norm-Equivalent Discrete Least-Squares Principles
.124
4.6
Concluding Remarks
.129
Contents xvii
Part III Least-Squares Finite Element Methods for Elliptic Problems
5
Scalar Elliptic Equations
.133
5.1
Applications of Scalar
Poisson
Equations
.135
5.2
Least-Squares Finite Element Methods for the Second-Order
Poisson
Equation
.137
5.2.1
Continuous Least-Squares Principles
.138
5.2.2
Discrete Least-Squares Principles
.139
5.3
First-Order System Reformulations
.140
5.3.1
The Div-Grad System
.141
5.3.2
The Extended Div-Grad System
.145
5.3.3
Application Examples
.146
5.4
Energy Balances
.147
5.4.1
Energy Balances in the Agmon-Douglis-Nirenberg
Setting
. 148
5.4.2
Energy Balances in the Vector-Operator Setting
. 152
5.5
Continuous Least-Squares Principles
. 159
5.6
Discrete Least-Squares Principles
. 163
5.6.1
The Div-Grad System
.163
5.6.2
The Extended Div-Grad System
.169
5.7
Error Analyses
.171
5.7.1
Error Estimates in Solution Space Norms
.171
5.7.2
L2(Q) Error Estimates
.175
5.8
Connections Between Compatible Least-Squares
and Standard Finite Element Methods
.176
5.8.1
The Compatible Least-Squares Finite Element Method
with a Reaction Term
.177
5.8.2
The Compatible Least-Squares Finite Element Method
Without a Reaction Term
.181
5.9
Practicality Issues
.182
5.9.1
Practical Rewards of Compatibility
.184
5.9.2
Compatible Least-Squares Finite Element Methods
on Non-Affine Grids
.190
5.9.3
Advantages and Disadvantages of Extended Systems
.192
5.10
A Summary of Conclusions and Recommendations
.194
6
Vector Elliptic Equations
.197
6.
1 Applications of Vector Elliptic Equations
.200
6.2
Reformulation of Vector Elliptic Problems
.201
6.2.
1 Div-Curl Systems
.202
6.2.2
Curl-Curl Systems
.203
6.3
Least-Squares Finite Element Methods for Div-Curl Systems
----206
6.3.1
Energy Balances
.206
6.3.2
Continuous Least-Squares Principles
.209
6.3.3
Discrete Least-Squares Principles
.211
xviii Contents
6.3.4
Analysis of Conforming Least-Squares Finite Element
Methods
.214
6.3.5
Analysis of Non-Conforming Least-Squares Finite
Element Methods
.216
6.4
Least-Squares Finite Element Methods for Curl-Curl Systems
. 221
6.4.1
Energy Balances
.221
6.4.2
Continuous Least-Squares Principles
.224
6.4.3
Discrete Least-Squares Principles
.225
6.4.4
Error Analysis
.230
6.5
Practicality Issues
.231
6.5.1
Solution of Algebraic Equations
.232
6.5.2
Implementation of Non-Conforming Methods
.234
6.6
A Summary of Conclusions
.236
7
The Stokes Equations
.237
7.1
First-Order System Formulations of the Stokes Equations
.238
7.1.1
The Velocity-Vorticity-Pressure System
.239
7.1.2
The Velocity-Stress-Pressure System
.242
7.1.3
The Velocity Gradient-Velocity-Pressure System
.243
7.2
Energy Balances in the Agmon-Douglis-Nirenberg Setting
.246
7.2.1
The Velocity-Vorticity-Pressure System
.247
7.2.2
The Velocity-Stress-Pressure System
.250
7.2.3
The Velocity Gradient-Velocity-Pressure System
.251
7.3
Continuous Least-Squares Principles
in the Agmon-Douglis-Nirenberg Setting
.253
7.3.1
The Velocity-Vorticity-Pressure System
.253
7.3.2
The Velocity-Stress-Pressure System
.256
7.3.3
The Velocity Gradient-Velocity-Pressure System
.256
7.4
Discrete Least-Squares Principles
in the Agmon-Douglis-Nirenberg Setting
.257
7.4.1
The Velocity-Vorticity-Pressure System
.258
7.4.2
The Velocity-Stress-Pressure System
.260
7.4.3
The Velocity Gradient-Velocity-Pressure System
.260
7.5
Error Estimates in the Agmon-Douglis-Nirenberg Setting
.261
7.5.1
The Velocity-Vorticity-Pressure System
.261
7.5.2
The Velocity-Stress-Pressure System
.263
7.5.3
The Velocity Gradient-Velocity-Pressure System
.264
7.6
Practicality Issues in the Agmon-Douglis-Nirenberg Setting
.264
7.6.1
Solution of the Discrete Equations
.265
7.6.2
Issues Related to Non-Homogeneous Elliptic Systems
. 266
7.6.3
Mass Conservation
.271
7.6.4
The Zero Mean Pressure Constraint
.274
7.7
Least-Squares Finite Element Methods
in the Vector-Operator Setting
.277
7.7.1
Energy Balances
.277
Contents xix
7.7.2
Continuous Least-Squares Principles
.281
7.7.3
Discrete Least-Squares Principles
.281
7.7.4
Stability of Discrete Least-Squares Principles
.284
7.7.5
Conservation of Mass and Strong Compatibility
.287
7.7.6
Error Estimates
.293
7.7.7
Connection Between Discrete Least-Squares Principles
and Mixed-Galerkin Methods
.302
7.7.8
Practicality Issues in the Vector Operator Setting
.304
7.8
A Summary of Conclusions and Recommendations
.306
Part IV Least-Squares Finite Element Methods for Other Settings
8
The Navier-Stokes Equations
.311
8.1
First-Order System Formulations of the Navier-Stokes Equations
.313
8.2
Least-Squares Principles for the Navier-Stokes Equations
.314
8.2.1
Continuous Least-Squares Principles
.315
8.2.2
Discrete Least-Squares Principles
.316
8.3
Analysis of Least-Squares Finite Element Methods
.317
8.3.1
Quotation of Background Results
.318
8.3.2
Compliant Discrete Least-Squares Principles
for the Velocity-Vorticity-Pressure System
.321
8.3.3
Norm-Equivalent Discrete Least-Squares Principles
for the Velocity-Vorticity-Pressure System
.329
8.3.4
Compliant Discrete Least-Squares Principles
for the Velocity Gradient-Velocity-Pressure System
.340
8.3.5
A Norm-Equivalent Discrete Least-Squares Principle
for the Velocity Gradient-Velocity-Pressure System
.344
8.4
Practicality Issues
.346
8.4.1
Solution of the Nonlinear Equations
.348
8.4.2
Implementation of Norm-Equivalent Methods
.351
8.4.3
The Utility of Discrete Negative Norm Least-Squares
Finite Element Methods
.354
8.4.4
Advantages and Disadvantages of Extended Systems
.359
8.5
A Summary of Conclusions and Recommendations
.364
9
Parabolic Partial Differential Equations
.367
9.1
The Generalized Heat Equation
.368
9.1.1
Backward-Euler Least-Squares Finite Element Methods
. 369
9.1.2
Second-Order Time Accurate Least-Squares Finite
Element Methods
.382
9.1.3
Comparison of Finite-Difference Least-Squares Finite
Element Methods
.389
9.1.4
Space-Time Least-Squares Principles
.391
9.1.5
Practical Issues
.395
9.2
The Time-Dependent Stokes Equations
.396
xx Contents
10
Hyperbolic Partial Differential Equations
.403
10.1
Model Conservation Law Problems
.404
10.2
Energy Balances
.406
10.2.1
Energy Balances in Hubert Spaces
.407
10.2.2
Energy Balances in Banach Spaces
.409
10.3
Continuous Least-Squares Principles
.410
10.3.1
Extension to Time-Dependent Conservation Laws
.412
10.4
Least-Squares Finite Element Methods in a Hubert Space Setting
.413
10.4.1
Conforming Methods
.413
10.4.2
Non-Conforming Methods
.414
10.5
Residual Minimization Methods in a Banach Space Setting
.416
10.5.1
An
Ľ(Í2)
Minimization Method
.416
10.5.2
Regularized L1
(β)
Minimization Method
.418
10.6
Least-Squares Finite Element Methods Based on Adaptively
Weighted
ί?(Ω)
Norms
.419
10.6.1
An Iteratively Re-Weighted Least-Squares Finite
Element Method
.419
10.6.2
A Feedback Least-Squares Finite Element Method
.420
10.7
Practicality Issues
.422
10.7.1
Approximation of Smooth Solutions
.422
10.7.2
Approximation of Discontinuous Solutions
.423
10.8
A Summary of Conclusions and Recommendations
.427
11
Control and Optimization Problems
.429
11.1
Quadratic Optimization and Control Problems in Hubert Spaces
with Linear Constraints
.431
11.1.1
Existence of Optimal States and Controls
.432
11.1.2
Least-Squares Formulation of the Constraint Equation
. 435
11.2
Solution via
Lagrange
Multipliers of the Optimal Control
Problem
.438
11.2.1
Galerkin Finite Element Methods for the Optimality
System
.439
11.2.2
Least-Squares Finite Element Methods for the Optimality
System
.442
11.3
Methods Based on Direct Penalization by the Least-Squares
Functional
.447
11.3.1
Discretization of the Perturbed Optimality System
.450
11.3.2
Discretization of the Eliminated System
.453
11.4
Methods Based on Constraining by the Least-Squares Functional
. 455
11.4.1
Discretization of the Optimality System
.457
11.4.2
Discretize-Then-Eliminate Approach for the Perturbed
Optimality System
.457
11.4.3
Eliminate-Then-Discretize Approach for the Perturbed
Optimality System
.459
11.5
Relative Merits of the Different Approaches
.460
11.6
Example: Optimization Problems for the Stokes Equations
.461
Contents xxi
11.6.1
The Optimization Problems and Galerkin Finite Element
Methods
.463
11.6.2
Least-Squares Finite Element Methods for the Constraint
Equations
.467
11.6.3
Least-Squares Finite Element Methods for the Optimality
Systems
.468
11.6.4
Constraining by the Least-Squares Functional
for the Constraint Equations
.471
12
Variations on Least-Squares Finite Element Methods
.475
12.1
Weak Enforcement of Boundary Conditions
.475
12.2
LL* Finite Element Methods
.480
12.3
Mimetic Reformulation of Least-Squares Finite Element Methods
. 483
12.4
Collocation Least-Squares Finite Element Methods
.488
12.5
Restricted Least-Squares Finite Element Methods
.490
12.6
Optimization-Based Least-Squares Finite Element Methods
.492
12.7
Least-Squares Finite Element Methods
for Advection-Diffusion-Reaction Problems
.494
12.8
Least-Squares Finite Element Methods for Higher-Order
Problems
.503
12.9
Least-Squares Finite Element Methods for Div-Grad-Curl
Systems
.505
12.10
Domain Decomposition Least-Squares Finite Element Methods.
. 507
12.11
Least-Squares Finite Element Methods for Multi-Physics
Problems
.513
12.12
Least-Squares Finite Element Methods for Problems
with Singular Solutions
.517
12.13
Treffetz Least-Squares Finite Element Methods
.521
12.14
A Posteriori Error Estimation and Adaptive Mesh Refinement
. 523
12.15
Least-Squares Wavelet Methods
.526
12.16
Meshless Least-Squares Methods
.528
Part V Supplementary Material
A Analysis Tools
.533
A.
1
General Notations and Symbols
.533
A.2 Function Spaces
.535
A.2.
1
The Sobolev Spaces
Η'{Ω)
.536
A.2.2 Spaces Related to the Gradient, Curl, and Divergence
Operators
.540
A.3 Properties of Function Spaces
.547
A.3.1 Embeddings of C(Q)
Γ)Ό(Ω)
.547
A.3.2
Poincaré-Friedrichs
Inequalities
.548
A.3.3 Hodge Decompositions
.550
A.3.4 Trace Theorems
.551
xxii Contents
В
Compatible
Finite Element
Spaces.553
B.I Formal Definition and Properties of Finite Element Spaces
.554
B.2 Finite Element Approximation of the
De Rham
Complex
.557
B.2.1
Examples of Compatible Finite Element Spaces
.559
B.2.2 Approximation of
C(ß)
ΠΌ(Ω)
.
567
B.2.
3
Exact Sequences of Finite Element Spaces
.569
B.3 Properties of Compatible Finite Element Spaces
.571
B.3.
1
Discrete Operators
.571
B.3.2 Discrete
Poincaré-Friedrichs
Inequalities
.576
B.3.3 Discrete Hodge Decompositions
.577
B.3.4 Inverse Inequalities
.580
B.4 Norm Approximations
.581
B.4.1 Quasi-Norm-Equivalent Approximations
.581
B.4.
2
Norm-Equivalent Approximations
.582
С
Linear Operator Equations in Hubert Spaces
.585
С
1
Auxiliary Operator Equations
.586
C.2 Energy Balances
.589
D
The Agmon-Douglis-Nirenberg Theory and Verifying
its Assumptions
.593
D.
1
The Agmon-Douglis-Nirenberg Theory
.593
D.2 Verifying the Assumptions of the Agmon-Douglis-Nirenberg
Theory
.597
D.2.1 Div-Grad Systems
.598
D.2.2 Div-Grad-Curl Systems
.602
D.2.3 Div-Curl Systems
.606
D.2.4 The Velocity-Vorticity-Pressure Formulation
of the Stokes System
.608
D.2.5 The Velocity-Stress-Pressure Formulation of the Stokes
System
.622
References
.625
Acronyms
.641
Glossary
.643
Index
.647
ISBN 978-0-387-30888-3
780387и308883и
The book examines theoretical and computational aspects of least-squares finite
element methods (LSFEMs) for partial differential equations (PDEs) arising in key
science and engineering applications. It is intended for mathematicians, scientists,
and engineers interested in either or both the theory and practice associated with the
numerical solution of PDEs.
The first part looks at strengths and weaknesses of classical
varìational
principles,
reviews alternative
varìational
formulations, and offers a glimpse at the main concepts
that enter into the formulation of LSFEMs. Subsequent parts introduce mathematical
frameworks for LSFEMs and their analysis, apply the frameworks to concrete PDEs,
and discuss computational properties of resulting LSFEMs. Abo included are recent
advances such as compatible LSFEMs, negative-norm LSFEMs, and LSFEMs for
optimal control and design problems. Numerical examples illustrate key aspects of
the theory ranging from the importance of norm-equivalence to connections between
compatible LSFEMs and classical-Galerkin and mixed-Galerkin methods.
Pavel Bochcv is a Distinguished Member of the Technical Staff at
Sandia
National
Laboratories with research interests in compatible discretizations for PDEs,
multiphysics problems, and scientific computing.
Max Gunzburger is Frances Eppes Professor of Scientific Computing and
Mathematics at Florida State University and recipient of the W.T. and Idelia Reid Prize
in Mathematics from the Society for Industrial and Applied Mathematics. |
| any_adam_object | 1 |
| author | Bochev, Pavel B. Gunzburger, Max D. |
| author_facet | Bochev, Pavel B. Gunzburger, Max D. |
| author_role | aut aut |
| author_sort | Bochev, Pavel B. |
| author_variant | p b b pb pbb m d g md mdg |
| building | Verbundindex |
| bvnumber | BV035436644 |
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| callnumber-label | QA1 |
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| callnumber-search | QA1 |
| callnumber-sort | QA 11 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 540 SK 910 SK 920 |
| ctrlnum | (OCoLC)76935728 (DE-599)DNB992207800 |
| 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 225 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035436644 |
| illustrated | Illustrated |
| indexdate | 2024-07-20T10:07:52Z |
| institution | BVB |
| isbn | 9780387308883 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017356958 |
| oclc_num | 76935728 |
| open_access_boolean | |
| owner | DE-20 DE-703 DE-29T DE-83 DE-188 DE-898 DE-BY-UBR DE-11 DE-824 |
| owner_facet | DE-20 DE-703 DE-29T DE-83 DE-188 DE-898 DE-BY-UBR DE-11 DE-824 |
| physical | XXII, 660 S. graph. Darst. |
| publishDate | 2009 |
| publishDateSearch | 2009 |
| publishDateSort | 2009 |
| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Bochev, Pavel B. Verfasser aut Least-squares finite element methods Pavel B. Bochev ; Max D. Gunzburger Least squares finite element methods New York, NY Springer 2009 XXII, 660 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 166 Differential equations, Partial Finite element method Least squares Methode der kleinsten Quadrate (DE-588)4038974-1 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 s Methode der kleinsten Quadrate (DE-588)4038974-1 s DE-604 Gunzburger, Max D. Verfasser aut Erscheint auch als Online-Ausgabe 978-0-387-68922-7 Applied mathematical sciences 166 (DE-604)BV000005274 166 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2718186&prov=M&dok_var=1&dok_ext=htm Inhaltstext Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017356958&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017356958&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Bochev, Pavel B. Gunzburger, Max D. Least-squares finite element methods Applied mathematical sciences Differential equations, Partial Finite element method Least squares Methode der kleinsten Quadrate (DE-588)4038974-1 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| subject_GND | (DE-588)4038974-1 (DE-588)4017233-8 |
| title | Least-squares finite element methods |
| title_alt | Least squares finite element methods |
| title_auth | Least-squares finite element methods |
| title_exact_search | Least-squares finite element methods |
| title_full | Least-squares finite element methods Pavel B. Bochev ; Max D. Gunzburger |
| title_fullStr | Least-squares finite element methods Pavel B. Bochev ; Max D. Gunzburger |
| title_full_unstemmed | Least-squares finite element methods Pavel B. Bochev ; Max D. Gunzburger |
| title_short | Least-squares finite element methods |
| title_sort | least squares finite element methods |
| topic | Differential equations, Partial Finite element method Least squares Methode der kleinsten Quadrate (DE-588)4038974-1 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| topic_facet | Differential equations, Partial Finite element method Least squares Methode der kleinsten Quadrate Finite-Elemente-Methode |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2718186&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=017356958&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017356958&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT bochevpavelb leastsquaresfiniteelementmethods AT gunzburgermaxd leastsquaresfiniteelementmethods |