Elliptic Differential Equations: Theory and Numerical Treatment
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Berlin
Springer Berlin
2010
|
| Edition: | 1st. softcover pr. |
| Series: | Springer series in computational mathematics
18 |
| Subjects: | |
| Online Access: | Inhaltstext Inhaltsverzeichnis |
| Item Description: | Aus dem Dt. |
| Physical Description: | XIII, 311 S. |
| ISBN: | 9783642052446 |
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Record in the Search Index
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| adam_text |
TABLE OF CONTENTS FOREWORD V TABLE OF CONTENTS VII NOTATION XILL 1
PARTIAL DIFFERENTIAL EQUATIONS AND THEIR CLASSIFICATION INTO TYPES 1 1.1
EXAMPLES 1 1.2 CLASSIFICATION OF SECOND-ORDER EQUATIONS INTO TYPES . 4
1.3 TYPE CLASSIFICATION FOR SYSTEMS OF FIRST ORDER . . . . 6 1.4
CHARACTERISTIC PROPERTIES OF THE DIFFERENT TYPES . . . 7 2 THE POTENTIAL
EQUATION 12 2.1 POSING THE PROBLEM 12 2.2 SINGULARITY FUNCTION 14 2.3
THE MEAN VALUE PROPERTY AND MAXIMUM PRINCIPLE . . 17 2.4 CONTINUOUS
DEPENDENCE ON THE BOUNDARY DATA . . . . 23 3 THE POISSON EQUATION 27 3.1
POSING THE PROBLEM 27 3.2 REPRESENTATION OF THE SOLUTION BY THE GREEN
FUNCTION . 28 3.3 THE GREEN FUNCTION FOR THE BALL 34 3.4 THE NEUMANN
BOUNDARY VALUE PROBLEM 35 3.5 THE INTEGRAL EQUATION METHOD 36 4
DIFFERENCE METHODS FOR THE POISSON EQUATION 38 4.1 INTRODUCTION: THE
ONE-DIMENSIONAL CASE 38 4.2 THE FIVE-POINT FORMULA 40 4.3 M-MATRICES,
MATRIX NORMS, POSITIVE DEFINITE MATRICES 44 44 PROPERTIES OF THE MATRI
BIBLIOGRAFISCHE INFORMATIONEN HTTP://D-NB.INFO/999137700 DIGITALISIERT
DURCH VIN TABLE OF CONTENTS 4.6 DISCRETISATIONS OF HIGHER ORDER 62 4.7
THE DISCRETISATION OF THE NEUMANN BOUNDARY VALUE PROBLEM 65 4.7.1
ONE-SIDED DIFFERENCE FOR DU/DN 65 4.7.2 SYMMETRIC DIFFERENCE FOR DU/DN
70 4.7.3 SYMMETRIC DIFFERENCE FOR DU/DN ON AN ONSET GRID 71 4.7.4 PROOF
OF THE STABILITY THEOREM 7 72 4.8 DISCRETISATION IN AN ARBITRARY DOMAIN
78 4.8.1 SHORTLEY-WEILER APPROXIMATION . 78 4.8.2 INTERPOLATION AT
POINTS NEAR THE BOUNDARY . . . 83 5 GENERAL BOUNDARY VALUE PROBLEMS 85
5.1 DIRICHLET BOUNDARY VALUE PROBLEMS FOR LINEAR DIFFERENTIAL EQUATIONS
85 5.1.1 POSING THE PROBLEM 85 5.1.2 MAXIMUM PRINCIPLE 86 5.1.3
UNIQUENESS OF THE SOLUTION AND CONTINUOUS DEPENDENCE 87 5.1.4 DIFFERENCE
METHODS FOR THE GENERAL DIFFERENTIAL EQUATION OF SECOND ORDER 90 5.1.5
GREEN'S FUNCTION 95 5.2 GENERAL BOUNDARY CONDITIONS 95 5.2.1 FORMULATING
THE BOUNDARY VALUE PROBLEM . . . 95 5.2.2 DIFFERENCE METHODS FOR GENERAL
BOUNDARY CONDITIONS 98 5.3 BOUNDARY PROBLEMS OF HIGHER ORDER 103 5.3.1
THE BIHARMONIC DIFFERENTIAL EQUATION 103 5.3.2 GENERAL LINEAR
DIFFERENTIAL EQUATIONS OF ORDER 2M 104 5.3.3 DISCRETISATION OF THE
BIHARMONIC DIFFERENTIAL EQUATIO TABLE OF CONTENTS DC 6.2.4 H*(TI) FOR
REAL S 0 122 6.2.5 TRACE AND EXTENSION THEOREMS 123 6.3 DUAL SPACES
130 6.3.1 DUAL SPACE OF A NORMED SPACE 130 6.3.2 ADJOINT OPERATORS 132
6.3.3 SCALES OF HUBERT SPACES 133 6.4 COMPACT OPERATORS 135 6.5 BILINEAR
FORMS 137 VARIATIONAL FORMULATION 144 7.1 HISTORICAL REMARKS 144 7.2
EQUATIONS WITH HOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS 145 7.3
INHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS . . . . 150 7.4 NATURAL
BOUNDARY CONDITIONS 152 THE METHOD OF FINITE ELEMENTS 161 8.1 THE
RITZ-GALERKIN METHOD 161 8.2 ERROR ESTIMATES 167 8.3 FINITE ELEMENTS 171
8.3.1 INTRODUCTION: LINEAR ELEMENTS FOR J? = (A, B) . . 171 8.3.2 LINEAR
ELEMENTS FOR UE C R A 174 8.3.3 BILINEAR ELEMENTS FOR SS C R 2 178 8.3.4
QUADRATIC ELEMENTS FOR SS C R 2 180 8.3.5 ELEMENTS FOR SS C R 3 182 8.3.6
HANDLING OF SIDE CONDITIONS 182 8.4 ERROR ESTIMATES FOR FINITE ELEMENT
METHODS 185 8.4.1 ^-ESTIMATES FOR LINEAR ELEMENTS 185 8.4.2 1? AN X
TABLE OF CONTENTS 8.7.2 THE TREFFTZ METHOD 200 8.7.3 FINITE-ELEMENT
METHODS FOR SINGULAR SOLUTIONS . 201 8.7.4 ADAPTIVE TRIANGULATION 201
8.7.5 HIERARCHICAL BASES 202 8.7.6 SUPERCONVERGENCE 202 8.8 PROPERTIES
OF THE STIFFNESS MATRIX 203 9 REGULARITY 208 9.1 SOLUTIONS OF THE
BOUNDARY VALUE PROBLEM IN H'{SI), A M 208 9.1.1 THE REGULARITY PROBLEM
208 9.1.2 REGULARITY THEOREMS FOR SS = R" 210 9.1.3 REGULARITY THEOREMS
FOR SS = R" 215 9.1.4 REGULARITY THEOREMS FOR GENERAL SS C ST . . . 219
9.1.5 REGULARITY FOR CONVEX DOMAINS AND DOMAINS WITH CORNERS 223 9.1.6
REGULARITY IN THE INTERIOR 226 9.2 REGULARITY PROPERTIES OF DIFFERENCE
EQUATIONS . . . . 226 9.2.1 DISCRETE ^-REGULARITY 226 9.2.2 CONSISTENCY
232 9.2.3 OPTIMAL ERROR ESTIMATES 238 9.2.4 A^-REGULARITY 240 10 SPECIAL
DIFFERENTIAL EQUATIONS 244 10.1 DIFFERENTIAL EQUATIONS WITH
DISCONTINUOUS COEFFICIENTS . 244 10.1.1 FORMULATION 244 10.1.2
DISCRETISATION 246 10.2 A SINGULAR PERTURBATION PROBLEM 247 10.2.1 THE
CONVECTION-DIFFUSION EQUATION 247 10.2.2 STABLE DIFFERENCE SCHEMES 249
10.2.3 FINITE ELEMENTS 251 11 EIGENVALUE PROBLEMS 253 11.1 FORMULATION
OF EIGENVALUE PROBLEMS 253 11.2 FINITE ELEMENT DISCRETISATION 254 11.2.1
DISCRETISATION 254 11.2. TABLE OF CONTENTS XI 12 STOKES EQUATIONS 275
12.1 SYSTEMS OF ELLIPTIC DIFFERENTIAL EQUATIONS 275 12.2 VARIATIONAL
FORMULATION 278 12.2.1 WEAK FORMULATION OF THE STOKES EQUATIONS . . .
278 12.2.2 SADDLEPOINT PROBLEMS 279 12.2.3 EXISTENCE AND UNIQUENESS OF
THE SOLUTION OF A SADDLEPOINT PROBLEM 282 12.2.4 SOLVABILITY AND
REGULARITY OF THE STOKES PROBLEM 285 12.2.5 A VO-ELLIPTIC VARIATIONAL
FORMULATION OF THE STOKES PROBLEM 289 12.3 MIXED FINITE-ELEMENT METHOD
FOR THE STOKES PROBLEM . 290 12.3.1 FINITE-ELEMENT DISCRETISATION OF A
SADDLEPOINT PROBLEM 290 12.3.2 STABILITY CONDITIONS 291 12.3.3 STABLE
FINITE-ELEMENT SPACES FOR THE STOKES PROBLEM 293 12.3.3.1 STABILITY
CRITERION 293 12.3.3.2 FINITE-ELEMENT DISCRETISATIONS WITH THE BUBBLE
FUNCTION 294 12.3.3.3 STABLE DISCRETISATIONS WITH LINEAR ELEMENTS IN V H
296 12.3.3.4 ERROR ESTIMATES 297 BIBLIOGRAPHY 300 INDEX 307 |
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| classification_rvk | SK 560 SK 920 |
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| dewey-full | 518.64 515.3533 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis 515 - Analysis |
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| dewey-search | 518.64 515.3533 |
| dewey-sort | 3518.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1st. softcover pr. |
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| spelling | Hackbusch, Wolfgang 1948- Verfasser (DE-588)115588582 aut Theorie und Numerik elliptischer Differentialgleichungen Elliptic Differential Equations Theory and Numerical Treatment Wolfgang Hackbusch 1st. softcover pr. Berlin Springer Berlin 2010 XIII, 311 S. txt rdacontent n rdamedia nc rdacarrier Springer series in computational mathematics 18 Aus dem Dt. Numerische Mathematik (DE-588)4042805-9 gnd rswk-swf Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf 1\p (DE-588)4123623-3 Lehrbuch gnd-content Elliptische Differentialgleichung (DE-588)4014485-9 s Numerisches Verfahren (DE-588)4128130-5 s DE-604 Numerische Mathematik (DE-588)4042805-9 s 2\p DE-604 Springer series in computational mathematics 18 (DE-604)BV000012004 18 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=3404707&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=020195331&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Hackbusch, Wolfgang 1948- Elliptic Differential Equations Theory and Numerical Treatment Springer series in computational mathematics Numerische Mathematik (DE-588)4042805-9 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4042805-9 (DE-588)4014485-9 (DE-588)4128130-5 (DE-588)4123623-3 |
| title | Elliptic Differential Equations Theory and Numerical Treatment |
| title_alt | Theorie und Numerik elliptischer Differentialgleichungen |
| title_auth | Elliptic Differential Equations Theory and Numerical Treatment |
| title_exact_search | Elliptic Differential Equations Theory and Numerical Treatment |
| title_full | Elliptic Differential Equations Theory and Numerical Treatment Wolfgang Hackbusch |
| title_fullStr | Elliptic Differential Equations Theory and Numerical Treatment Wolfgang Hackbusch |
| title_full_unstemmed | Elliptic Differential Equations Theory and Numerical Treatment Wolfgang Hackbusch |
| title_short | Elliptic Differential Equations |
| title_sort | elliptic differential equations theory and numerical treatment |
| title_sub | Theory and Numerical Treatment |
| topic | Numerische Mathematik (DE-588)4042805-9 gnd Elliptische Differentialgleichung (DE-588)4014485-9 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Numerische Mathematik Elliptische Differentialgleichung Numerisches Verfahren Lehrbuch |
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