Finite difference methods on irregular networks: a generalized approach to second order elliptic problems
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
1987
|
| Series: | International series of numerical mathematics
82 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | 206 S. |
| ISBN: | 3764318805 0817618805 |
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| 100 | 1 | |a Heinrich, Bernd |d 1943-2020 |e Verfasser |0 (DE-588)106591346X |4 aut | |
| 245 | 1 | 0 | |a Finite difference methods on irregular networks |b a generalized approach to second order elliptic problems |c Bernd Heinrich |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 1987 | |
| 300 | |a 206 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a International series of numerical mathematics |v 82 | |
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| 650 | 0 | 7 | |a Finite-Differenzen-Methode |0 (DE-588)4194626-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Differenzenverfahren |0 (DE-588)4134362-1 |2 gnd |9 rswk-swf |
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Record in the Search Index
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| adam_text | CONTENTS
1. INTRODUCTION 9
1.1. Preliminary remarks 9
1.2. Scope of monograph 11
1.3. Plan of monograph, comments 12
2. BOUNDARY VALUE PROBLEMS AND IRREGULAR NETWORKS 17
2.1. A class of elliptic problems 17
2.2. Irregular networks 23
2.2.1. Networks of triangles and rectangles 23
2.2.2. Locally irregular networks 27
2.3. Secondary networks and boxes 29
2.3.1. General remarks 29
2.3.2. Secondary networks via the perpendicular bisectors (PB) 30
2.3.3. Secondary networks via the medians (MD) 36
3. CONSTRUCTION OP FINITE DIFFERENCE APPROXIMATIONS 40
3.1. The principle of approximation 40
3.2. Finite difference schemes via method (PB) 43
3.2.1. The approximation of the balance equations 43
3.2.2. Finite difference schemes of the type (PB) 47
3.3. Finite difference schemes via method (MD) 48
3.3 1. Quadrature formulas for the balance equations 48
3.3.2. Finite difference quotients 50
3.3.3. Finite difference schemes of the type (MD) 52
4. ANALYTICAL AND MATRIX PROPERTIES OF THE DIFFERENCE 55
OPERATORS Ah
4.1. General remarks and notations 55
4.2. Monotonicity and other matrix properties 57
4.2.1. Operators A. and matrices via method (PB) 57
4.2.2. MD operators Ah in comparison with PB operators 60
4.3. Scalar products, norms and a trace theorem 64
4.3.1. Notations for scalar products and norms 64
4.3.2. a f^ equivalence for norms of special functions 67
4.3.3. F r^ relations for norms of special functions 70
4.3.4. Relations between continuous and discrete norms 72
4.3.5. A trace theorem and the equivalence of certain grid norms 73
6
4.4. Green s formula, inequalities of Friedrichs Polncare 75
type and the positive definiteness of A.
4.4.1. Green s formula, the symmetry and the energy of A, 75
n
4.4.2. Inequalities of Frledrichs Poincare type 78
4.4.3. The positive definiteness of Ah 80
4.5. A priori estimates for A, using the wl and C norm 85
4.5.1. A priori estimates using the wl norm 85
4.5.2. Weak imbedding of WjCuJ) in C(t5) 88
4.5.3. A priori estimates using the C norm 90
5. ERROR ESTIMATES AND CONVERGENCE 96
5.1. Error splitting and approaches to the error estimation 96
5.1.1 . Introductory remarks and error splitting for PB schemes 96
5.1.2. Two kinds of error splitting for MD schemes 99
5.1.3. A priori estimates of the error 102
5.1.4. Convergence for classical solutions of C (fl) type v.l 2) 106
5.1.5. Convergence for generalized solutions of tfi(fl.) type (l»2)108
5.2. The error ae of the principal part of PB operators 109
5.2.1. The splitting of ae and first estimates 109
5.2.2. The choice of local estimation regions 112
5.2.3. The estimation of Xy. and atg 114
5.2.4. The estimation of *u 119
5.2.5. The weakening of the assumptions and conclusions 122
5.3. The error ae of the principal part of MD operators 124
5.3.1. Splittings and estimates via centres of gravity 124
5.3.2. Splittings and estimates via mesh midpoints 127
5.4. The error y^ for PB and MD schemes 132
5.4.1. ^ estimates for grid points x £ CO 132
5.4.2. Yt[~estima tes for Srid points x€fl 2j 135
5.4.3. The modification and weakening of some assumptions 142
5.5. Convergence for W?(Q) solutions 146
5.5.1. Convergence in the discrete Wg norm 146
5.5.2. Convergence in the discrete C norm 149
6. FINITE DIFFERENCE SCHEMES FOR NONSYMMETRIC PROBLEMS 150
6.1. Construction of finite difference approximations 150
6.1.1. Boundary value problem and balance equations 150
6.1.2. Upwind difference quotients 152
6.1.3. Further difference quotients and the FDSs A*Jy = Fh 154
6.2. Properties of the difference operators A? 158
6.2.1. Monotonicity and a priori estimates 158
6.2.2. Positive definiteness and a priori estimates 161
7
6.3. The error ^ of the convection term 162
6.3.1. Error splitting and estimates 162
6.3.2. The approach to estimating Yr for ueWgCft) (1 2) 164
6.3.3. Local and global bounds of f^ 166
6.4. Convergence for C1(H) and W^ClD solutions (1»2) 172
7. CONCLUDING REMARKS 174
APPENDICES 181
1. Appendix DI: Relations of Differential and Integral 181
calculus, norms
2. Appendix ES: Estimation of functionals on Sobolev spaces 183
3. Appendix EX: Extension of functions 185
4. Appendix SE: Some relations of geometry 186
5. Appendix IM: Imbedding and trace theorems 187
6. Appendix TR: Affine transformations of coordinates and 190
functionals
REFERENCES 194
SUBJECT INDEX 203
LIST OF FIGURES 205
ABBREVIATIONS 205
NOTATIONS 206
8
|
| any_adam_object | 1 |
| author | Heinrich, Bernd 1943-2020 |
| author_GND | (DE-588)106591346X |
| author_facet | Heinrich, Bernd 1943-2020 |
| author_role | aut |
| author_sort | Heinrich, Bernd 1943-2020 |
| author_variant | b h bh |
| building | Verbundindex |
| bvnumber | BV000802406 |
| classification_rvk | SK 910 SK 920 SK 980 |
| classification_tum | MAT 674f |
| ctrlnum | (OCoLC)230925839 (DE-599)BVBBV000802406 |
| dewey-full | 515.62 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.62 |
| dewey-search | 515.62 |
| dewey-sort | 3515.62 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| illustrated | Not Illustrated |
| indexdate | 2024-07-09T15:19:41Z |
| institution | BVB |
| isbn | 3764318805 0817618805 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-000502743 |
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| physical | 206 S. |
| publishDate | 1987 |
| publishDateSearch | 1987 |
| publishDateSort | 1987 |
| publisher | Birkhäuser |
| record_format | marc |
| series | International series of numerical mathematics |
| series2 | International series of numerical mathematics |
| spelling | Heinrich, Bernd 1943-2020 Verfasser (DE-588)106591346X aut Finite difference methods on irregular networks a generalized approach to second order elliptic problems Bernd Heinrich Basel [u.a.] Birkhäuser 1987 206 S. txt rdacontent n rdamedia nc rdacarrier International series of numerical mathematics 82 Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Randwertproblem (DE-588)4048395-2 gnd rswk-swf Finite-Differenzen-Methode (DE-588)4194626-1 gnd rswk-swf Differenzenverfahren (DE-588)4134362-1 gnd rswk-swf Elliptisches Randwertproblem (DE-588)4193399-0 gnd rswk-swf Finite-Differenzen-Methode (DE-588)4194626-1 s Randwertproblem (DE-588)4048395-2 s Elliptische Differentialgleichung (DE-588)4014485-9 s DE-604 Elliptisches Randwertproblem (DE-588)4193399-0 s Differenzenverfahren (DE-588)4134362-1 s International series of numerical mathematics 82 (DE-604)BV035415862 82 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000502743&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Heinrich, Bernd 1943-2020 Finite difference methods on irregular networks a generalized approach to second order elliptic problems International series of numerical mathematics Elliptische Differentialgleichung (DE-588)4014485-9 gnd Randwertproblem (DE-588)4048395-2 gnd Finite-Differenzen-Methode (DE-588)4194626-1 gnd Differenzenverfahren (DE-588)4134362-1 gnd Elliptisches Randwertproblem (DE-588)4193399-0 gnd |
| subject_GND | (DE-588)4014485-9 (DE-588)4048395-2 (DE-588)4194626-1 (DE-588)4134362-1 (DE-588)4193399-0 |
| title | Finite difference methods on irregular networks a generalized approach to second order elliptic problems |
| title_auth | Finite difference methods on irregular networks a generalized approach to second order elliptic problems |
| title_exact_search | Finite difference methods on irregular networks a generalized approach to second order elliptic problems |
| title_full | Finite difference methods on irregular networks a generalized approach to second order elliptic problems Bernd Heinrich |
| title_fullStr | Finite difference methods on irregular networks a generalized approach to second order elliptic problems Bernd Heinrich |
| title_full_unstemmed | Finite difference methods on irregular networks a generalized approach to second order elliptic problems Bernd Heinrich |
| title_short | Finite difference methods on irregular networks |
| title_sort | finite difference methods on irregular networks a generalized approach to second order elliptic problems |
| title_sub | a generalized approach to second order elliptic problems |
| topic | Elliptische Differentialgleichung (DE-588)4014485-9 gnd Randwertproblem (DE-588)4048395-2 gnd Finite-Differenzen-Methode (DE-588)4194626-1 gnd Differenzenverfahren (DE-588)4134362-1 gnd Elliptisches Randwertproblem (DE-588)4193399-0 gnd |
| topic_facet | Elliptische Differentialgleichung Randwertproblem Finite-Differenzen-Methode Differenzenverfahren Elliptisches Randwertproblem |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=000502743&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV035415862 |
| work_keys_str_mv | AT heinrichbernd finitedifferencemethodsonirregularnetworksageneralizedapproachtosecondorderellipticproblems |