Understanding and implementing the finite element method:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
SIAM
2006
|
| Series: | Other titles in applied mathematics
97 |
| Subjects: | |
| Online Access: | Table of contents only Publisher description Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references (p. 353-355) and index |
| Physical Description: | XVI, 363 S. graph. Darst. 26 cm |
| ISBN: | 0898716144 9780898716146 |
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MARC
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| 084 | |a MAT 674f |2 stub | ||
| 100 | 1 | |a Gockenbach, Mark S. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Understanding and implementing the finite element method |c Mark S. Gockenbach |
| 264 | 1 | |a Philadelphia |b SIAM |c 2006 | |
| 300 | |a XVI, 363 S. |b graph. Darst. |c 26 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Other titles in applied mathematics |v 97 | |
| 500 | |a Includes bibliographical references (p. 353-355) and index | ||
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Finite element method | |
| 650 | 4 | |a Finite element method |x Data processing | |
| 650 | 0 | 7 | |a MATLAB |0 (DE-588)4329066-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Finite-Elemente-Methode |0 (DE-588)4017233-8 |2 gnd |9 rswk-swf |
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| 689 | 0 | 1 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |D s |
| 689 | 0 | |5 DE-604 | |
| 689 | 1 | 0 | |a Finite-Elemente-Methode |0 (DE-588)4017233-8 |D s |
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| 689 | 1 | |5 DE-604 | |
| 830 | 0 | |a Other titles in applied mathematics |v 97 |w (DE-604)BV023088396 |9 97 | |
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| 856 | 4 | |u http://www.loc.gov/catdir/enhancements/fy0708/2006045012-d.html |3 Publisher description | |
| 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=015591567&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-015591567 | ||
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| _version_ | 1804136439490805760 |
|---|---|
| adam_text | Contents
Preface
xiii
I The Basic Framework for Stationary Problems
1
1
Some model PDEs
3
1.1
Laplace s equation; elliptic BVPs
.................... 3
1.1.1
Physical experiments modeled by Laplace s equation
... 5
1.2
Other elliptic BVPs
........................... 8
1.2.1
The equations of
isotropie
elasticity
............ 8
1.2.2
General linear elasticity
.................. 10
1.3
Exercises for Chapter
1 ......................... 11
2
The weak form of a BVP
15
2.1
Review of vector calculus
........................ 15
2.1.1
The divergence theorem
.................. 15
2.1.2
Green s identity
...................... 17
2.1.3
Other forms of the divergence theorem and Green s identity
18
2.2
The weak form of a BVP
........................ 20
2.2.1
Minimization of energy
.................. 21
2.2.2
Relaxing the PDE
..................... 23
2.2.3
A few details about Sobolev spaces
............ 27
2.3
The weak form for other boundary conditions and PDEs
........ 29
2.3.1
Neumann conditions and the weak form
......... 29
2.3.2
Mixed boundary conditions
................ 31
2.3.3
Inhomogeneous boundary conditions
........... 31
2.3.4
Other elliptic BVPs
.................... 33
2.4
Existence and uniqueness theory for the weak form of a BVP
..... 35
2.4.1
Vector spaces and inner products
............. 35
2.4.2
Hubert spaces
....................... 39
2.4.3
Linear functionals
..................... 41
2.4.4
The Riesz representation theorem
............. 42
2.4.5
Variational problems and the Riesz representation theorem
42
vii
v¡¡¡
____________ Contents
2.5
Examples of ellipticity
..........................45
2.5.1
The model problem
.................... 45
2.5.2
The equations of
isotropie
elasticity
............ 48
2.6
Variational formulation of nonsymmetric problems
........... 51
2.7
Exercises for Chapter
2......................... 53
3
The Galerkin method 57
3.1
The projection theorem
......................... 57
3.2
The Galerkin method for a variational problem
............. 59
3.2.1
Another interpretation of the Galerkin method
...... 62
3.2.2
The Galerkin method for a nonsymmetric problem
.... 63
3.3
Exercises for Chapter
3......................... 63
4
Piecewise polynomials and the finite element method
67
4.1
Piecewise linear functions defined on a triangular mesh
........ 67
4.1.1
Using piecewise linear functions in Galerkin s method
. . 70
4.1.2
The sparsity of the stiffness matrix
............ 74
4.2
Quadratic
Lagrange
triangles
...................... 77
4.2.1
Continuous piecewise quadratic functions
......... 77
4.2.2
The finite element method with quadratic
Lagrange
triangles
78
4.3
Cubic
Lagrange
triangles
........................ 80
4.3.1
Continuous piecewise cubic functions
........... 80
4.3.2
The finite element method with cubic
Lagrange
triangles
. 83
4.4 Lagrange
triangles of arbitrary degree
.................. 84
4.4.1
Hierarchical bases for finite element spaces
........ 85
4.5
Other finite elements: Rectangles and quadrilaterals
.......... 86
4.5.1
Rectangular elements
................... 86
4.5.2
General quadrilaterals
................... 87
4.6
Using a reference triangle in finite element calculations
........ 91
4.7
Isoparametric finite element methods
.................. 93
4.7.1
Isoparametric quadratic triangles
............. 96
4.7.2
Isoparametric triangles of higher degree
.......... 100
4.8
Exercises for Chapter
4......................... 101
S
Convergence of the finite element method
105
5.1
Approximating smooth functions by continuous piecewise linear func¬
tions
...................................105
5.1.1
The standard refinement of
a triangulation
........106
5.1.2
Nondegenerate
fomüies
of
triangulations
.........106
5.1.3
Approximation by piecewise linear functions
.......107
5.2
Approximation by higher-order piecewise polynomials
.........108
5.3
Convergence in the energy norm
....................
HO
5.4
Convergence in the L2-norm
......................115
5.5
Variational crimes
............................
llg
5.5.1
Numerical integration
...................118
Contents ix
5.5.2 Outline
of the analysis of the effect of
quadrature
.....120
5.5.3
Isoparametric finite elements
...............121
5.6
Exercises for Chapter
5.........................122
II Data Structures and Implementation
125
6
The mesh data structure
127
6.1
Programming the finite element method
.................127
6.1.1
Assembling the stiffness matrix
..............127
6.1.2
Computing the load vector
.................131
6.2
The mesh data structure
.........................134
6.2.1
The list of nodes
......................134
6.2.2
The list of edges
......................135
6.2.3
The list of elements
....................135
6.2.4
The list of free boundary edges
..............137
6.2.5
Other fields in the mesh data structure
...........137
6.3
The
MATLAB
implementation
.....................138
6.3.1
Generating a mesh by refinement
.............139
6.3.2
Generating a mesh from a triangle-node list
........140
6.3.3
Assessing the quality of
a triangulation
..........142
6.3.4
Viewing a mesh
......................144
6.3.5
Handling a domain with a curved boundary
........147
6.3.6
Viewing a piecewise linear function
............148
6.3.7
MATLAB
functions
....................150
6.3.8
A summary of the notation
.................151
6.4
Exercises for Chapter
6.........................152
7
Programming the finite element method: Linear
Lagrange
triangles
155
7.1
Quadrature
................................ 155
7.1.1
Gaussian quadrature
.................... 155
7.1.2
Evaluating the standard basis functions on a triangle
... 162
7.1.3
Quadrature over a square
................. 165
7.2
Assembling the stiffness matrix
..................... 166
7.3
Computing the load vector
........................ 168
7.3.1
Inhomogeneous Dirichlet conditions
........... 169
7.3.2
Inhomogeneous Neumann conditions
........... 170
7.4
Examples
................................. 171
7.4.1
Homogeneous boundary conditions
............ 173
7.4.2
Inhomogeneous boundary conditions
........... 174
7.4.3
A more realistic example
................. 179
7.5
The
MATLAB
implementation
..................... 182
7.5.1
MATLAB
functions
.................... 182
7.6
Exercises for Chapter
7......................... 183
Contents
Lagrange
triangles
of arbitrary degree
187
8.1
Quadrature for higher-order elements
..................187
8.2
Assembling the stiffness matrix and load vector
............192
8.3
Implementing the isoparametric method
................195
8.3.1
Placement of nodes in the isoparametric method
.....199
8.4
Examples
.................................200
8.5
The
MATLAB
implementation
.....................203
8.5.1
version2
........................203
8.5.2
version3
........................205
8.6
Exercises for Chapter
8.........................206
The finite element method for general BVPs
209
9.1
Scalar BVPs
...............................209
9.1.1
An example
........................212
9.2 Isotropie
elasticity
............................213
9.3
Mesh locking
..............................218
9.4
The
MATLAB
implementation
.....................220
9.5
Exercises for Chapter
9.........................221
III Solving the Finite Element Equations
223
10
Direct solution of sparse linear systems
225
10.1
The Cholesky factorization for positive definite matrices
........225
10.1.1
The Cholesky factorization for dense matrices
......226
10.1.2
The Cholesky factorization for banded matrices
.....228
10.2
Factoring general sparse matrices
....................229
10.3
Exercises for Chapter
10.........................233
11
Iterative methods: Conjugate gradients
235
11.1
The
CG
method
.............................235
11.1.1
The
CG
algorithm
.....................239
11.1.2
Convergence of the
CG
algorithm
.............243
11.2
Hierarchical bases for finite element spaces
...............244
11.2.1
Hierarchical bases for linear
Lagrange
triangles
.....244
11.2.2
Relationship between the stiffness matrices in nodal and
hierarchical bases
.....................249
11.3
The hierarchical basis
CG
method
....................250
11.4
The preconditioned
CG
method
.....................252
11.4.1
Alternate derivation of PCG
................253
11.4.2
Preconditioned
......................254
11.5
The pure Neumann problem
.......................256
11.6
The
MATLAB
implementation
.....................262
H
.6.1
MATLAB
functions
....................262
11.7
Exercises for Chapter
11.........................263
Contents xi
12
The classical stationary iterations
267
12.1
Stationary iterations
...........................267
12.1.1
Matrix norms
........................268
12.1.2
Convergence of stationary iterations
............270
12.2
The classical iterations
..........................270
12.2.1
Jacobi iteration
.......................271
12.2.2
Gauss-Seidel iteration
...................272
12.2.3
SOR
iteration
.......................273
12.2.4
Symmetric
SOR
......................274
12.2.5 CG
with SSOR preconditioning
..............275
12.3
The
MATLAB
implementation
.....................276
12.3.1
MATLAB
functions
....................276
12.4
Exercises for Chapter
12.........................276
13
The
m u
It ¡grid method
279
13.1
Stationary iterations as smoothers
....................279
13.1.1
The stiffness matrix for the model problem
........279
13.1.2
Fourier modes and the spectral decomposition of
К
. . .281
13.1.3
Jacobi iteration
.......................284
13.1.4
Weighted Jacobi iteration
.................287
13.2
The coarse grid correction algorithm
..................291
13.2.1
Projecting the equation onto a coarser mesh
........292
13.2.2
The projected equation and the Galerkin idea
.......294
13.2.3
The two-grid multigrid algorithm
.............295
13.3
The multigrid V-cycle
..........................296
13.3.1
W-cycles and ^-cycles
...................300
13.4
Full multigrid
..............................300
13.4.1
Discretization, algebraic, and total errors
.........303
13.5
The
MATLAB
implementation
.....................304
13.5.1
MATLAB
functions
....................304
13.6
Exercises for Chapter
13.........................305
IV Adaptive Methods
307
14
Adaptive mesh generation
309
14.1
Algorithms for local mesh refinement
..................311
14.1.1
Algorithms based on the standard refinement
.......311
14.1.2
Algorithms based on bisection
...............312
14.2
Selecting triangles for local refinement
.................315
14.3
A complete adaptive algorithm
.....................317
14.4
The
MATLAB
implementation
.....................322
14.4,!
MATLAB
functions
....................324
14.5
Exercises for Chapter
14.........................325
x¡¡
Contents
15
Error estimators and indicators
329
15.1
An explicit error indicator based on estimating the curvature of the
solution
.................................330
15.2
An explicit error indicator based on the residual
............334
15.3
The element residual error estimator
..................340
15.4
Some final examples
...........................345
15.4.1
A discontinuous coefficient
................346
15.4.2
A reentrant corner
.....................346
15.4.3
Transition from Dirichlet to Neumann conditions
.....348
15.5
The
MATLAB
implementation
.....................349
15.5.1
MATLAB
functions
....................349
15.6
Exercises for Chapter
15.........................349
Bibliography
353
Index
357
|
| adam_txt |
Contents
Preface
xiii
I The Basic Framework for Stationary Problems
1
1
Some model PDEs
3
1.1
Laplace's equation; elliptic BVPs
. 3
1.1.1
Physical experiments modeled by Laplace's equation
. 5
1.2
Other elliptic BVPs
. 8
1.2.1
The equations of
isotropie
elasticity
. 8
1.2.2
General linear elasticity
. 10
1.3
Exercises for Chapter
1 . 11
2
The weak form of a BVP
15
2.1
Review of vector calculus
. 15
2.1.1
The divergence theorem
. 15
2.1.2
Green's identity
. 17
2.1.3
Other forms of the divergence theorem and Green's identity
18
2.2
The weak form of a BVP
. 20
2.2.1
Minimization of energy
. 21
2.2.2
Relaxing the PDE
. 23
2.2.3
A few details about Sobolev spaces
. 27
2.3
The weak form for other boundary conditions and PDEs
. 29
2.3.1
Neumann conditions and the weak form
. 29
2.3.2
Mixed boundary conditions
. 31
2.3.3
Inhomogeneous boundary conditions
. 31
2.3.4
Other elliptic BVPs
. 33
2.4
Existence and uniqueness theory for the weak form of a BVP
. 35
2.4.1
Vector spaces and inner products
. 35
2.4.2
Hubert spaces
. 39
2.4.3
Linear functionals
. 41
2.4.4
The Riesz representation theorem
. 42
2.4.5
Variational problems and the Riesz representation theorem
42
vii
v¡¡¡
_ Contents
2.5
Examples of ellipticity
.45
2.5.1
The model problem
. 45
2.5.2
The equations of
isotropie
elasticity
. 48
2.6
Variational formulation of nonsymmetric problems
. 51
2.7
Exercises for Chapter
2. 53
3
The Galerkin method 57
3.1
The projection theorem
. 57
3.2
The Galerkin method for a variational problem
. 59
3.2.1
Another interpretation of the Galerkin method
. 62
3.2.2
The Galerkin method for a nonsymmetric problem
. 63
3.3
Exercises for Chapter
3. 63
4
Piecewise polynomials and the finite element method
67
4.1
Piecewise linear functions defined on a triangular mesh
. 67
4.1.1
Using piecewise linear functions in Galerkin's method
. . 70
4.1.2
The sparsity of the stiffness matrix
. 74
4.2
Quadratic
Lagrange
triangles
. 77
4.2.1
Continuous piecewise quadratic functions
. 77
4.2.2
The finite element method with quadratic
Lagrange
triangles
78
4.3
Cubic
Lagrange
triangles
. 80
4.3.1
Continuous piecewise cubic functions
. 80
4.3.2
The finite element method with cubic
Lagrange
triangles
. 83
4.4 Lagrange
triangles of arbitrary degree
. 84
4.4.1
Hierarchical bases for finite element spaces
. 85
4.5
Other finite elements: Rectangles and quadrilaterals
. 86
4.5.1
Rectangular elements
. 86
4.5.2
General quadrilaterals
. 87
4.6
Using a reference triangle in finite element calculations
. 91
4.7
Isoparametric finite element methods
. 93
4.7.1
Isoparametric quadratic triangles
. 96
4.7.2
Isoparametric triangles of higher degree
. 100
4.8
Exercises for Chapter
4. 101
S
Convergence of the finite element method
105
5.1
Approximating smooth functions by continuous piecewise linear func¬
tions
.105
5.1.1
The standard refinement of
a triangulation
.106
5.1.2
Nondegenerate
fomüies
of
triangulations
.106
5.1.3
Approximation by piecewise linear functions
.107
5.2
Approximation by higher-order piecewise polynomials
.108
5.3
Convergence in the energy norm
.
HO
5.4
Convergence in the L2-norm
.115
5.5
Variational crimes
.
llg
5.5.1
Numerical integration
.118
Contents ix
5.5.2 Outline
of the analysis of the effect of
quadrature
.120
5.5.3
Isoparametric finite elements
.121
5.6
Exercises for Chapter
5.122
II Data Structures and Implementation
125
6
The mesh data structure
127
6.1
Programming the finite element method
.127
6.1.1
Assembling the stiffness matrix
.127
6.1.2
Computing the load vector
.131
6.2
The mesh data structure
.134
6.2.1
The list of nodes
.134
6.2.2
The list of edges
.135
6.2.3
The list of elements
.135
6.2.4
The list of free boundary edges
.137
6.2.5
Other fields in the mesh data structure
.137
6.3
The
MATLAB
implementation
.138
6.3.1
Generating a mesh by refinement
.139
6.3.2
Generating a mesh from a triangle-node list
.140
6.3.3
Assessing the quality of
a triangulation
.142
6.3.4
Viewing a mesh
.144
6.3.5
Handling a domain with a curved boundary
.147
6.3.6
Viewing a piecewise linear function
.148
6.3.7
MATLAB
functions
.150
6.3.8
A summary of the notation
.151
6.4
Exercises for Chapter
6.152
7
Programming the finite element method: Linear
Lagrange
triangles
155
7.1
Quadrature
. 155
7.1.1
Gaussian quadrature
. 155
7.1.2
Evaluating the standard basis functions on a triangle
. 162
7.1.3
Quadrature over a square
. 165
7.2
Assembling the stiffness matrix
. 166
7.3
Computing the load vector
. 168
7.3.1
Inhomogeneous Dirichlet conditions
. 169
7.3.2
Inhomogeneous Neumann conditions
. 170
7.4
Examples
. 171
7.4.1
Homogeneous boundary conditions
. 173
7.4.2
Inhomogeneous boundary conditions
. 174
7.4.3
A more realistic example
. 179
7.5
The
MATLAB
implementation
. 182
7.5.1
MATLAB
functions
. 182
7.6
Exercises for Chapter
7. 183
Contents
Lagrange
triangles
of arbitrary degree
187
8.1
Quadrature for higher-order elements
.187
8.2
Assembling the stiffness matrix and load vector
.192
8.3
Implementing the isoparametric method
.195
8.3.1
Placement of nodes in the isoparametric method
.199
8.4
Examples
.200
8.5
The
MATLAB
implementation
.203
8.5.1
version2
.203
8.5.2
version3
.205
8.6
Exercises for Chapter
8.206
The finite element method for general BVPs
209
9.1
Scalar BVPs
.209
9.1.1
An example
.212
9.2 Isotropie
elasticity
.213
9.3
Mesh locking
.218
9.4
The
MATLAB
implementation
.220
9.5
Exercises for Chapter
9.221
III Solving the Finite Element Equations
223
10
Direct solution of sparse linear systems
225
10.1
The Cholesky factorization for positive definite matrices
.225
10.1.1
The Cholesky factorization for dense matrices
.226
10.1.2
The Cholesky factorization for banded matrices
.228
10.2
Factoring general sparse matrices
.229
10.3
Exercises for Chapter
10.233
11
Iterative methods: Conjugate gradients
235
11.1
The
CG
method
.235
11.1.1
The
CG
algorithm
.239
11.1.2
Convergence of the
CG
algorithm
.243
11.2
Hierarchical bases for finite element spaces
.244
11.2.1
Hierarchical bases for linear
Lagrange
triangles
.244
11.2.2
Relationship between the stiffness matrices in nodal and
hierarchical bases
.249
11.3
The hierarchical basis
CG
method
.250
11.4
The preconditioned
CG
method
.252
11.4.1
Alternate derivation of PCG
.253
11.4.2
Preconditioned
.254
11.5
The pure Neumann problem
.256
11.6
The
MATLAB
implementation
.262
H
.6.1
MATLAB
functions
.262
11.7
Exercises for Chapter
11.263
Contents xi
12
The classical stationary iterations
267
12.1
Stationary iterations
.267
12.1.1
Matrix norms
.268
12.1.2
Convergence of stationary iterations
.270
12.2
The classical iterations
.270
12.2.1
Jacobi iteration
.271
12.2.2
Gauss-Seidel iteration
.272
12.2.3
SOR
iteration
.273
12.2.4
Symmetric
SOR
.274
12.2.5 CG
with SSOR preconditioning
.275
12.3
The
MATLAB
implementation
.276
12.3.1
MATLAB
functions
.276
12.4
Exercises for Chapter
12.276
13
The
m u
It ¡grid method
279
13.1
Stationary iterations as smoothers
.279
13.1.1
The stiffness matrix for the model problem
.279
13.1.2
Fourier modes and the spectral decomposition of
К
. . .281
13.1.3
Jacobi iteration
.284
13.1.4
Weighted Jacobi iteration
.287
13.2
The coarse grid correction algorithm
.291
13.2.1
Projecting the equation onto a coarser mesh
.292
13.2.2
The projected equation and the Galerkin idea
.294
13.2.3
The two-grid multigrid algorithm
.295
13.3
The multigrid V-cycle
.296
13.3.1
W-cycles and ^-cycles
.300
13.4
Full multigrid
.300
13.4.1
Discretization, algebraic, and total errors
.303
13.5
The
MATLAB
implementation
.304
13.5.1
MATLAB
functions
.304
13.6
Exercises for Chapter
13.305
IV Adaptive Methods
307
14
Adaptive mesh generation
309
14.1
Algorithms for local mesh refinement
.311
14.1.1
Algorithms based on the standard refinement
.311
14.1.2
Algorithms based on bisection
.312
14.2
Selecting triangles for local refinement
.315
14.3
A complete adaptive algorithm
.317
14.4
The
MATLAB
implementation
.322
14.4,!
MATLAB
functions
.324
14.5
Exercises for Chapter
14.325
x¡¡
Contents
15
Error estimators and indicators
329
15.1
An explicit error indicator based on estimating the curvature of the
solution
.330
15.2
An explicit error indicator based on the residual
.334
15.3
The element residual error estimator
.340
15.4
Some final examples
.345
15.4.1
A discontinuous coefficient
.346
15.4.2
A reentrant corner
.346
15.4.3
Transition from Dirichlet to Neumann conditions
.348
15.5
The
MATLAB
implementation
.349
15.5.1
MATLAB
functions
.349
15.6
Exercises for Chapter
15.349
Bibliography
353
Index
357 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Gockenbach, Mark S. |
| author_facet | Gockenbach, Mark S. |
| author_role | aut |
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| dewey-full | 518/.25 |
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| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518/.25 |
| dewey-search | 518/.25 |
| dewey-sort | 3518 225 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
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| illustrated | Illustrated |
| index_date | 2024-07-02T17:11:51Z |
| indexdate | 2024-07-09T20:56:25Z |
| institution | BVB |
| isbn | 0898716144 9780898716146 |
| language | English |
| lccn | 2006045012 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015591567 |
| oclc_num | 266438288 |
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| physical | XVI, 363 S. graph. Darst. 26 cm |
| publishDate | 2006 |
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| series2 | Other titles in applied mathematics |
| spelling | Gockenbach, Mark S. Verfasser aut Understanding and implementing the finite element method Mark S. Gockenbach Philadelphia SIAM 2006 XVI, 363 S. graph. Darst. 26 cm txt rdacontent n rdamedia nc rdacarrier Other titles in applied mathematics 97 Includes bibliographical references (p. 353-355) and index Datenverarbeitung Finite element method Finite element method Data processing MATLAB (DE-588)4329066-8 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 gnd rswk-swf Finite-Elemente-Methode (DE-588)4017233-8 s Partielle Differentialgleichung (DE-588)4044779-0 s DE-604 MATLAB (DE-588)4329066-8 s Other titles in applied mathematics 97 (DE-604)BV023088396 97 http://www.loc.gov/catdir/toc/fy0703/2006045012.html Table of contents only http://www.loc.gov/catdir/enhancements/fy0708/2006045012-d.html Publisher description Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015591567&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gockenbach, Mark S. Understanding and implementing the finite element method Other titles in applied mathematics Datenverarbeitung Finite element method Finite element method Data processing MATLAB (DE-588)4329066-8 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| subject_GND | (DE-588)4329066-8 (DE-588)4044779-0 (DE-588)4017233-8 |
| title | Understanding and implementing the finite element method |
| title_auth | Understanding and implementing the finite element method |
| title_exact_search | Understanding and implementing the finite element method |
| title_exact_search_txtP | Understanding and implementing the finite element method |
| title_full | Understanding and implementing the finite element method Mark S. Gockenbach |
| title_fullStr | Understanding and implementing the finite element method Mark S. Gockenbach |
| title_full_unstemmed | Understanding and implementing the finite element method Mark S. Gockenbach |
| title_short | Understanding and implementing the finite element method |
| title_sort | understanding and implementing the finite element method |
| topic | Datenverarbeitung Finite element method Finite element method Data processing MATLAB (DE-588)4329066-8 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Finite-Elemente-Methode (DE-588)4017233-8 gnd |
| topic_facet | Datenverarbeitung Finite element method Finite element method Data processing MATLAB Partielle Differentialgleichung Finite-Elemente-Methode |
| url | http://www.loc.gov/catdir/toc/fy0703/2006045012.html http://www.loc.gov/catdir/enhancements/fy0708/2006045012-d.html http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015591567&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV023088396 |
| work_keys_str_mv | AT gockenbachmarks understandingandimplementingthefiniteelementmethod |