Discontinuous Galerkin methods for solving elliptic and parabolic equations: theory and implementation
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Philadelphia
Society for Industrial and Applied Mathematics
[2008]
|
| Series: | Frontiers in applied mathematics
35 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | xxii, 190 Seiten Illustrationen, Diagramme |
| ISBN: | 9780898716566 |
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MARC
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| 020 | |a 9780898716566 |9 978-0-898716-56-6 | ||
| 035 | |a (OCoLC)226292048 | ||
| 035 | |a (DE-599)BVBBV023407391 | ||
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| 100 | 1 | |a Rivière, Béatrice |e Verfasser |0 (DE-588)1198878738 |4 aut | |
| 245 | 1 | 0 | |a Discontinuous Galerkin methods for solving elliptic and parabolic equations |b theory and implementation |c Béatrice Rivière |
| 264 | 1 | |a Philadelphia |b Society for Industrial and Applied Mathematics |c [2008] | |
| 300 | |a xxii, 190 Seiten |b Illustrationen, Diagramme | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Frontiers in applied mathematics |v 35 | |
| 500 | |a Includes bibliographical references and index | ||
| 650 | 7 | |a Análise numérica |2 larpcal | |
| 650 | 7 | |a Equações diferenciais parciais elíticas-parabólicas quasilineares |2 larpcal | |
| 650 | 7 | |a Método dos elementos finitos |2 larpcal | |
| 650 | 4 | |a Differential equations, Elliptic |x Numerical solutions | |
| 650 | 4 | |a Differential equations, Parabolic |x Numerical solutions | |
| 650 | 4 | |a Galerkin methods | |
| 650 | 0 | 7 | |a Parabolische Differentialgleichung |0 (DE-588)4173245-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Elliptische Differentialgleichung |0 (DE-588)4014485-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Diskontinuierliche Galerkin-Methode |0 (DE-588)4588309-9 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Diskontinuierliche Galerkin-Methode |0 (DE-588)4588309-9 |D s |
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| 830 | 0 | |a Frontiers in applied mathematics |v 35 |w (DE-604)BV001873790 |9 35 | |
| 856 | 4 | 2 | |m Digitalisierung UB Bayreuth |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016590062&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 999 | |a oai:aleph.bib-bvb.de:BVB01-016590062 | ||
Record in the Search Index
| _version_ | 1804137793270579200 |
|---|---|
| adam_text | Contents
List of
Figures
xv
List of Tables
xvii
List of Algorithms
xix
Preface
xxi
I Elliptic Problems
1
1
One-dimensional problem
3
1.1
Model problem
.............................. 3
1.2
A class of DG methods
......................... 3
1.3
Existence and uniqueness of the
ЕЮ
solution
.............. 6
1.4
Linear system
.............................. 7
1.4.1
Computing the matrix A
.................. 8
1.4.2
Computing the right-hand side
b
............. 10
1.4.3
Imposing boundary conditions strongly
.......... 11
1.5
Convergence of the DG method
..................... 12
1.6
Numerical experiments
......................... 13
1.7
Bibliographical remarks
......................... 15
Exercises
..................................... 17
2
Higher dimensional problem
19
2.1
Preliminaries
............................... 19
2.1.1
Vector notation
....................... 19
2.1.2
Sobolev spaces
....................... 19
2.1.3
Trace theorems
....................... 22
2.1.4
Approximation properties
................. 24
2.1.5
Green s theorem
...................... 24
2.1.6
Cauchy-Schwarz s and Young s inequalities
....... 25
2.2
Model problem
.............................. 25
2.2.1
Weak solution
....................... 26
2.2.2
Numerical solution
..................... 26
ix
Contents
2.3
Broken
Sobolev
spaces
......................... 27
2.3.1 Jumps
and averages
.................... 28
2.4
Variatíona]
formulation
......................... 29
2.4.1
Consistency
........................ 30
2.5 Finite element spaces
.......................... 32
2.5.1
Reference elements versus physical elements
....... 32
2.5.2
Basis functions
....................... 35
2.5.3
Numerical quadrature
................... 36
2.6
DG scheme
................................ 37
2.7
Properties
................................ 38
2.7.1
Coercivity of bilinear forms
................ 38
2.7.2
Continuity of bilinear form
................ 40
2.7.3
Local mass conservation
.................. 41
2.7.4
Existence and uniqueness of DG solution
......... 42
2.8
Error analysis
.............................. 42
2.8.1
Error estimates in the energy norm
............ 42
2.8.2
Error estimates in the L2 norm
............... 46
2.9
Implementing the DG method
...................... 49
2.9.1
Data structure
....................... 49
2.9.2
Local matrices and right-hand sides
............ 51
2.9.3
Global matrix and right-hand side
............. 55
2.10
Numerical experiments
......................... 57
2.10.1
Smooth solution
...................... 57
2.10.2
Singular solution
...................... 58
2.10.3
Condition number
..................... 59
2.11
The local discontinuous Galerkin method
................ 59
2.11.1
Definition of the mixed DG method
............ 60
2.11.2
Existence and uniqueness of the solution
......... 62
2.11.3
A priori error estimates
................... 63
2.12
DG versus classical finite element method
............... 64
2.13
Bibliographical remarks
......................... 67
Exercises
..................................... 67
Π
Parabolic Problems
69
3
Purely parabolic problems
71
3.1
Preliminaries
............................... 71
3.1.1
Functional spaces
..................... 71
3.1.2
Gronwalľs
inequalities
.................. 71
3.1.3
Taylor s expansions
.................... 72
3.1.4
Poincaré s
inequalities
................... 72
3.1.5
Inverse inequalities
.................... 73
3.2
Model problem
.............................. 73
3.3
Semidiscrete formulation
........................ 73
3.3.1
Apriori
bounds
....................... 75
Contents
x¡
3.3.2
Error estimates
.......................77
3.4
Fully discrete formulation
........................80
3.4.1
Backward
Euler
discretization
............... 80
3.4.2
Forward
Euler
discretization
................ 84
3.4.3
Crank-Nicolson discretization
............... 86
3.4.4
Runge-Kutta discretization
................ 87
3.4.5
DG in time discretization
................. 88
3.5
Implementation
............................. 91
3.6
Bibliographical remarks
......................... 92
Exercises
..................................... 92
4
Parabolic problems with convection
95
4.1
Model problem
..............................95
4.2
Semidiscrete formulation
........................96
4.2.1
Existence and uniqueness of solution
...........97
4.2.2
Consistency
........................97
4.2.3
Error estimates
.......................98
4.3
Fully discrete formulation
........................100
4.3.1
Overshoot and undershoot
.................100
4.3.2
Slope
limiters
.......................101
4.3.3
An improved
DG
method
.................104
4.4
Bibliographical remarks
.........................106
Exercises
.....................................106
Ш
Applications
107
5
Linear elasticity
109
5.1
Preliminaries
...............................109
5.1.1
Strain and stress tensors
..................109
5.1.2
Korn s inequalities
.....................110
5.2
Model problem
..............................110
5.3
DG scheme
................................
Ill
5.3.1
Consistency
........................112
5.3.2
Local equilibrium
.....................112
5.3.3
Coercivity
.........................112
5.4
Error analysis
..............................113
5.5
Bibliographical remarks
.........................115
Exercises
.....................................115
6
Stokes flow
117
6.1
Preliminaries
...............................117
6.1.1
Vector notation
.......................117
6.1.2
Barycentric coordinates
..................117
6.1.3
An approximation operator of degree one
.........119
6.1.4
An approximation operator of higher degree
.......120
xii__________________________________________________ Contents
6.1.5
Local
L2
projection
....................121
6.1.6
General
inf
-sup
condition
.................121
6.2
Model
problem
and weak
solution
...................122
6.3
DGscheme
................................123
6.3.1
Existence and uniqueness of solution
...........124
6.3.2
Local mass conservation
..................124
6.4
Discrete inf-sup condition
........................125
6.5
Error estimates
..............................126
6.6
Numerical results
............................129
6.7
Bibliographical remarks
.........................129
Exercises
.....................................130
7
Navier-Stokesflow
131
7.1
Preliminaries
...............................131
7.1.1
Sobolev imbedding
....................131
7.1.2
Holder s inequality.
....................132
7.1.3
Brouwer s fixed point theorem
..............132
7.2
Model problem and weak solution
...................133
7.3
DG discretization
............................133
7.3.1
Nonlinear convective term
.................134
7.3.2
Scheme
...........................136
7.3.3
Consistency
........................136
7.4
Existence and uniqueness of solution
..................136
7.4.1
Existence of discrete velocity
...............136
7.4.2
Existence of discrete pressure
...............137
7.4.3
Apriori
bounds
.......................138
7.4.4
Uniqueness
.........................138
7.5
A priori error estimates
.........................139
7.6
Numerical experiments
.........................140
7.6.1
Effects of penalty size
...................140
7.6.2
Step channel problem
...................141
7.7
Bibliographical remarks
.........................143
8
Flow in porous media
145
8.1
Two-phase flow
.............................145
8.1.1 Modelproblem.......................146
8.1.2
A sequential approach
...................148
8.1.3
A coupled approach
....................150
8.1.4
Numerical examples
....................151
8.2
Miscible
displacement
..........................153
8.2.1
Semidiscrete formulation
.................155
8.2.2
A fully discrete approach
.................156
8.2.3
Numerical examples
....................157
8.3
Bibliographical remarks
.........................158
Contents xiii
A Quadrature rules
159
A.1 Gauss quadrature rale on intervals
...................159
A.2 Quadrature rales on the reference triangle
................159
A.3 Quadrature rale on the reference quadrilateral
.............161
В
DG
codes
163
B.I
A
MATLAB
implementation for a one-dimensional problem
......163
B.2 Selected
С
routines for higher dimensional problem
..........165
С
An approximation result
175
Bibliography
179
Index
189
|
| adam_txt |
Contents
List of
Figures
xv
List of Tables
xvii
List of Algorithms
xix
Preface
xxi
I Elliptic Problems
1
1
One-dimensional problem
3
1.1
Model problem
. 3
1.2
A class of DG methods
. 3
1.3
Existence and uniqueness of the
ЕЮ
solution
. 6
1.4
Linear system
. 7
1.4.1
Computing the matrix A
. 8
1.4.2
Computing the right-hand side
b
. 10
1.4.3
Imposing boundary conditions strongly
. 11
1.5
Convergence of the DG method
. 12
1.6
Numerical experiments
. 13
1.7
Bibliographical remarks
. 15
Exercises
. 17
2
Higher dimensional problem
19
2.1
Preliminaries
. 19
2.1.1
Vector notation
. 19
2.1.2
Sobolev spaces
. 19
2.1.3
Trace theorems
. 22
2.1.4
Approximation properties
. 24
2.1.5
Green's theorem
. 24
2.1.6
Cauchy-Schwarz's and Young's inequalities
. 25
2.2
Model problem
. 25
2.2.1
Weak solution
. 26
2.2.2
Numerical solution
. 26
ix
Contents
2.3
Broken
Sobolev
spaces
. 27
2.3.1 Jumps
and averages
. 28
2.4
Variatíona]
formulation
. 29
2.4.1
Consistency
. 30
2.5 Finite element spaces
. 32
2.5.1
Reference elements versus physical elements
. 32
2.5.2
Basis functions
. 35
2.5.3
Numerical quadrature
. 36
2.6
DG scheme
. 37
2.7
Properties
. 38
2.7.1
Coercivity of bilinear forms
. 38
2.7.2
Continuity of bilinear form
. 40
2.7.3
Local mass conservation
. 41
2.7.4
Existence and uniqueness of DG solution
. 42
2.8
Error analysis
. 42
2.8.1
Error estimates in the energy norm
. 42
2.8.2
Error estimates in the L2 norm
. 46
2.9
Implementing the DG method
. 49
2.9.1
Data structure
. 49
2.9.2
Local matrices and right-hand sides
. 51
2.9.3
Global matrix and right-hand side
. 55
2.10
Numerical experiments
. 57
2.10.1
Smooth solution
. 57
2.10.2
Singular solution
. 58
2.10.3
Condition number
. 59
2.11
The local discontinuous Galerkin method
. 59
2.11.1
Definition of the mixed DG method
. 60
2.11.2
Existence and uniqueness of the solution
. 62
2.11.3
A priori error estimates
. 63
2.12
DG versus classical finite element method
. 64
2.13
Bibliographical remarks
. 67
Exercises
. 67
Π
Parabolic Problems
69
3
Purely parabolic problems
71
3.1
Preliminaries
. 71
3.1.1
Functional spaces
. 71
3.1.2
Gronwalľs
inequalities
. 71
3.1.3
Taylor's expansions
. 72
3.1.4
Poincaré's
inequalities
. 72
3.1.5
Inverse inequalities
. 73
3.2
Model problem
. 73
3.3
Semidiscrete formulation
. 73
3.3.1
Apriori
bounds
. 75
Contents
x¡
3.3.2
Error estimates
.77
3.4
Fully discrete formulation
.80
3.4.1
Backward
Euler
discretization
. 80
3.4.2
Forward
Euler
discretization
. 84
3.4.3
Crank-Nicolson discretization
. 86
3.4.4
Runge-Kutta discretization
. 87
3.4.5
DG in time discretization
. 88
3.5
Implementation
. 91
3.6
Bibliographical remarks
. 92
Exercises
. 92
4
Parabolic problems with convection
95
4.1
Model problem
.95
4.2
Semidiscrete formulation
.96
4.2.1
Existence and uniqueness of solution
.97
4.2.2
Consistency
.97
4.2.3
Error estimates
.98
4.3
Fully discrete formulation
.100
4.3.1
Overshoot and undershoot
.100
4.3.2
Slope
limiters
.101
4.3.3
An improved
DG
method
.104
4.4
Bibliographical remarks
.106
Exercises
.106
Ш
Applications
107
5
Linear elasticity
109
5.1
Preliminaries
.109
5.1.1
Strain and stress tensors
.109
5.1.2
Korn's inequalities
.110
5.2
Model problem
.110
5.3
DG scheme
.
Ill
5.3.1
Consistency
.112
5.3.2
Local equilibrium
.112
5.3.3
Coercivity
.112
5.4
Error analysis
.113
5.5
Bibliographical remarks
.115
Exercises
.115
6
Stokes flow
117
6.1
Preliminaries
.117
6.1.1
Vector notation
.117
6.1.2
Barycentric coordinates
.117
6.1.3
An approximation operator of degree one
.119
6.1.4
An approximation operator of higher degree
.120
xii_ Contents
6.1.5
Local
L2
projection
.121
6.1.6
General
inf
-sup
condition
.121
6.2
Model
problem
and weak
solution
.122
6.3
DGscheme
.123
6.3.1
Existence and uniqueness of solution
.124
6.3.2
Local mass conservation
.124
6.4
Discrete inf-sup condition
.125
6.5
Error estimates
.126
6.6
Numerical results
.129
6.7
Bibliographical remarks
.129
Exercises
.130
7
Navier-Stokesflow
131
7.1
Preliminaries
.131
7.1.1
Sobolev imbedding
.131
7.1.2
Holder's inequality.
.132
7.1.3
Brouwer's fixed point theorem
.132
7.2
Model problem and weak solution
.133
7.3
DG discretization
.133
7.3.1
Nonlinear convective term
.134
7.3.2
Scheme
.136
7.3.3
Consistency
.136
7.4
Existence and uniqueness of solution
.136
7.4.1
Existence of discrete velocity
.136
7.4.2
Existence of discrete pressure
.137
7.4.3
Apriori
bounds
.138
7.4.4
Uniqueness
.138
7.5
A priori error estimates
.139
7.6
Numerical experiments
.140
7.6.1
Effects of penalty size
.140
7.6.2
Step channel problem
.141
7.7
Bibliographical remarks
.143
8
Flow in porous media
145
8.1
Two-phase flow
.145
8.1.1 Modelproblem.146
8.1.2
A sequential approach
.148
8.1.3
A coupled approach
.150
8.1.4
Numerical examples
.151
8.2
Miscible
displacement
.153
8.2.1
Semidiscrete formulation
.155
8.2.2
A fully discrete approach
.156
8.2.3
Numerical examples
.157
8.3
Bibliographical remarks
.158
Contents xiii
A Quadrature rules
159
A.1 Gauss quadrature rale on intervals
.159
A.2 Quadrature rales on the reference triangle
.159
A.3 Quadrature rale on the reference quadrilateral
.161
В
DG
codes
163
B.I
A
MATLAB
implementation for a one-dimensional problem
.163
B.2 Selected
С
routines for higher dimensional problem
.165
С
An approximation result
175
Bibliography
179
Index
189 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Rivière, Béatrice |
| author_GND | (DE-588)1198878738 |
| author_facet | Rivière, Béatrice |
| author_role | aut |
| author_sort | Rivière, Béatrice |
| author_variant | b r br |
| building | Verbundindex |
| bvnumber | BV023407391 |
| callnumber-first | Q - Science |
| callnumber-label | QA377 |
| callnumber-raw | QA377 |
| callnumber-search | QA377 |
| callnumber-sort | QA 3377 |
| callnumber-subject | QA - Mathematics |
| classification_rvk | SK 920 |
| ctrlnum | (OCoLC)226292048 (DE-599)BVBBV023407391 |
| dewey-full | 518/.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518/.64 |
| dewey-search | 518/.64 |
| dewey-sort | 3518 264 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| discipline_str_mv | Mathematik |
| format | Book |
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| id | DE-604.BV023407391 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:26:06Z |
| indexdate | 2024-07-09T21:17:56Z |
| institution | BVB |
| isbn | 9780898716566 |
| language | English |
| lccn | 2008018508 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016590062 |
| oclc_num | 226292048 |
| open_access_boolean | |
| owner | DE-703 DE-706 DE-29T DE-20 DE-83 DE-11 DE-384 |
| owner_facet | DE-703 DE-706 DE-29T DE-20 DE-83 DE-11 DE-384 |
| physical | xxii, 190 Seiten Illustrationen, Diagramme |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | marc |
| series | Frontiers in applied mathematics |
| series2 | Frontiers in applied mathematics |
| spelling | Rivière, Béatrice Verfasser (DE-588)1198878738 aut Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation Béatrice Rivière Philadelphia Society for Industrial and Applied Mathematics [2008] xxii, 190 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Frontiers in applied mathematics 35 Includes bibliographical references and index Análise numérica larpcal Equações diferenciais parciais elíticas-parabólicas quasilineares larpcal Método dos elementos finitos larpcal Differential equations, Elliptic Numerical solutions Differential equations, Parabolic Numerical solutions Galerkin methods Parabolische Differentialgleichung (DE-588)4173245-5 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd rswk-swf Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 s Elliptische Differentialgleichung (DE-588)4014485-9 s Parabolische Differentialgleichung (DE-588)4173245-5 s DE-604 Frontiers in applied mathematics 35 (DE-604)BV001873790 35 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016590062&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Rivière, Béatrice Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation Frontiers in applied mathematics Análise numérica larpcal Equações diferenciais parciais elíticas-parabólicas quasilineares larpcal Método dos elementos finitos larpcal Differential equations, Elliptic Numerical solutions Differential equations, Parabolic Numerical solutions Galerkin methods Parabolische Differentialgleichung (DE-588)4173245-5 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd |
| subject_GND | (DE-588)4173245-5 (DE-588)4014485-9 (DE-588)4588309-9 |
| title | Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation |
| title_auth | Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation |
| title_exact_search | Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation |
| title_exact_search_txtP | Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation |
| title_full | Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation Béatrice Rivière |
| title_fullStr | Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation Béatrice Rivière |
| title_full_unstemmed | Discontinuous Galerkin methods for solving elliptic and parabolic equations theory and implementation Béatrice Rivière |
| title_short | Discontinuous Galerkin methods for solving elliptic and parabolic equations |
| title_sort | discontinuous galerkin methods for solving elliptic and parabolic equations theory and implementation |
| title_sub | theory and implementation |
| topic | Análise numérica larpcal Equações diferenciais parciais elíticas-parabólicas quasilineares larpcal Método dos elementos finitos larpcal Differential equations, Elliptic Numerical solutions Differential equations, Parabolic Numerical solutions Galerkin methods Parabolische Differentialgleichung (DE-588)4173245-5 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Diskontinuierliche Galerkin-Methode (DE-588)4588309-9 gnd |
| topic_facet | Análise numérica Equações diferenciais parciais elíticas-parabólicas quasilineares Método dos elementos finitos Differential equations, Elliptic Numerical solutions Differential equations, Parabolic Numerical solutions Galerkin methods Parabolische Differentialgleichung Elliptische Differentialgleichung Diskontinuierliche Galerkin-Methode |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016590062&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV001873790 |
| work_keys_str_mv | AT rivierebeatrice discontinuousgalerkinmethodsforsolvingellipticandparabolicequationstheoryandimplementation |