Accuracy Verification Methods: Theory and Algorithms
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| Main Author: | |
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| Format: | Electronic eBook |
| Language: | English |
| Published: |
2014
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| Series: | Computational Methods in Applied Sciences
32 |
| Subjects: | |
| Online Access: | BTU01 FHA01 FHI01 FHN01 FHR01 FKE01 FRO01 FWS01 FWS02 UBY01 Volltext Inhaltsverzeichnis Abstract |
| Item Description: | The importance of accuracy verification methods was understood at the very beginning of the development of numerical analysis. Recent decades have seen a rapid growth of results related to adaptive numerical methods and a posteriori estimates. However, in this important area there often exists a noticeable gap between mathematicians creating the theory and researchers developing applied algorithms that could be used in engineering and scientific computations for guaranteed and efficient error control. The goals of the book are to (1) give a transparent explanation of the underlying mathematical theory in a style accessible not only to advanced numerical analysts but also to engineers and students; (2) present detailed step-by-step algorithms that follow from a theory; (3) discuss their advantages and drawbacks, areas of applicability, give recommendations and examples |
| Physical Description: | 1 Online-Ressource (XIII, 355 p.) 75 illus |
| ISBN: | 9789400775817 |
| DOI: | 10.1007/978-94-007-7581-7 |
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| 245 | 1 | 0 | |a Accuracy Verification Methods |b Theory and Algorithms |c by Olli Mali, Pekka Neittaanmäki, Sergey Repin |
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| 490 | 0 | |a Computational Methods in Applied Sciences |v 32 | |
| 500 | |a The importance of accuracy verification methods was understood at the very beginning of the development of numerical analysis. Recent decades have seen a rapid growth of results related to adaptive numerical methods and a posteriori estimates. However, in this important area there often exists a noticeable gap between mathematicians creating the theory and researchers developing applied algorithms that could be used in engineering and scientific computations for guaranteed and efficient error control. The goals of the book are to (1) give a transparent explanation of the underlying mathematical theory in a style accessible not only to advanced numerical analysts but also to engineers and students; (2) present detailed step-by-step algorithms that follow from a theory; (3) discuss their advantages and drawbacks, areas of applicability, give recommendations and examples | ||
| 505 | 0 | |a 1 Errors Arising In Computer Simulation Methods -- 1.1 General scheme -- 1.2 Errors of mathematical models -- 1.3 Approximation errors -- 1.4 Numerical errors -- 2 Error Indicators -- 2.1 Error indicators and adaptive numerical methods -- 2.1.1 Error indicators for FEM solutions -- 2.1.2 Accuracy of error indicators -- 2.2 Error indicators for the energy norm -- 2.2.1 Error indicators based on interpolation estimates -- 2.2.2 Error indicators based on approximation of the error functional -- 2.2.3 Error indicators of the Runge type -- 2.3 Error indicators for goal-oriented quantities -- 2.3.1 Error indicators relying on the superconvergence of averaged fluxes in the primal and adjoint problems -- 2.3.2 Error indicators using the superconvergence of approximations in the primal problem -- 2.3.3 Error indicators based on partial equilibration of fluxes in the original problem -- 3 Guaranteed Error Bounds I -- 3.1 Ordinary differential equations -- | |
| 505 | 0 | |a 3.1.1 Derivation of guaranteed error bounds -- 3.1.2 Computation of error bounds -- 3.2 Partial differential equations -- 3.2.1 Maximal deviation from the exact solution -- 3.2.2 Minimal deviation from the exact solution -- 3.2.3 Particular cases -- 3.2.4 Problems with mixed boundary conditions -- 3.2.5 Estimates of global constants entering the majorant -- 3.2.6 Error majorants based on Poincar'e inequalities -- 3.2.7 Estimates with partially equilibrated fluxes -- 3.3 Error control algorithms -- 3.3.1 Global minimization of the majorant -- 3.3.2 Getting an error bound by local procedures -- 3.4 Indicators based on error majorants -- 3.5 Applications to adaptive methods -- 3.6 Combined (primal-dual) error norms and the majorant -- 4 Guaranteed Error Bounds II -- 4.1 Linear elasticity -- 4.1.1 Introduction -- 4.1.2 Euler–Bernoulli beam -- 4.1.3 The Kirchhoff–Love arch model -- 4.1.4 The Kirchhoff–Love plate -- 4.1.5 The Reissner–Mindlin plate -- 4.1.6 3D linear elasticity -- | |
| 505 | 0 | |a 4.1.7 The plane stress model -- 4.1.8 The plane strain model -- 4.2 The Stokes Problem -- 4.2.1 Divergence-free approximations -- 4.2.2 Approximations with nonzero divergence -- 4.2.3 Stokes problem in rotating system -- 4.3 A simple Maxwell type problem -- 4.3.1 Estimates of deviations from exact solutions -- 4.3.2 Numerical examples -- 4.4 Generalizations -- 4.4.1 Error majorant -- 4.4.2 Error minorant -- 5 Errors Generated By Uncertain Data -- 5.1 Mathematical models with incompletely known data -- 5.2 The accuracy limit -- 5.3 Estimates of the worst and best case scenario errors -- 5.4 Two-sided bounds of the radius of the solution set -- 5.5 Computable estimates of the radius of the solution set -- 5.5.1 Using the majorant -- 5.5.2 Using a reference solution -- 5.5.3 An advanced lower bound -- 5.6 Multiple sources of indeterminacy -- 5.6.1 Incompletely known right-hand side -- 5.6.2 The reaction diffusion problem -- 5.7 Error indication and indeterminate data -- | |
| 505 | 0 | |a 5.8 Linear elasticity with incompletely known Poisson ratio -- 5.8.1 Sensitivity of the energy functional -- 5.8.2 Example: axisymmetric model -- 6 Overview Of Other Results And Open Problems -- 6.1 Error estimates for approximations violating conformity -- 6.2 Linear elliptic equations -- 6.3 Time-dependent problems -- 6.4 Optimal control and inverse problems -- 6.5 Nonlinear boundary value problems -- 6.5.1 Variational inequalities -- 6.5.2 Elastoplasticity -- 6.5.3 Problems with power growth energy functionals -- 6.6 Modeling errors -- 6.7 Error bounds for iteration methods -- 6.7.1 General iteration algorithm -- 6.7.2 A priori estimates of errors -- 6.7.3 A posteriori estimates of errors -- 6.7.4 Advanced forms of error bounds -- 6.7.5 Systems of linear simultaneous equations -- 6.7.6 Ordinary differential equations -- 6.8 Roundoff errors -- 6.9 Open problems -- A Mathematical Background -- A.1 Vectors and tensors -- A.2 Spaces of functions -- A.2.1 Lebesgue and Sobolev spaces -- | |
| 505 | 0 | |a A.2.2 Boundary traces -- A.2.3 Linear functionals -- A.3 Inequalities -- A.3.1 The Hölder inequality -- A.3.2 The Poincaré and Friedrichs inequalities -- A.3.3 Korn’s inequality -- A.3.4 LBB inequality -- A.4 Convex functionals -- B Boundary Value Problems -- B.1 Generalized solutions of boundary value problems -- B.2 Variational statements of elliptic boundary value problems -- B.3 Saddle point statements of elliptic boundary value problems -- B.3.1 Introduction to the theory of saddle points -- B.3.2 Saddle point statements of linear elliptic problems -- B.3.3 Saddle point statements of nonlinear variational problems -- B.4 Numerical methods -- B.4.1 Finite difference methods -- B.4.2 Variational difference methods -- B.4.3 Petrov–Galerkin methods -- B.4.4 Mixed finite element methods -- B.4.5 Trefftz methods -- B.4.6 Finite volume methods -- B.4.7 Discontinuous Galerkin methods -- B.4.8 Fictitious domain methods -- C A Priori Verification Of Accuracy -- | |
| 505 | 0 | |a C.1 Projection error estimate -- C.2 Interpolation theory in Sobolev spaces -- C.3 A priori convergence rate estimates -- C.4 A priori error estimates for mixed FEM -- References -- Notation -- Index | |
| 650 | 4 | |a Computer science | |
| 650 | 4 | |a Electronic data processing | |
| 650 | 4 | |a Numerical analysis | |
| 650 | 4 | |a Engineering | |
| 650 | 4 | |a Computer Science | |
| 650 | 4 | |a Numeric Computing | |
| 650 | 4 | |a Computational Science and Engineering | |
| 650 | 4 | |a Numerical Analysis | |
| 650 | 4 | |a Numerical and Computational Physics | |
| 650 | 4 | |a Computational Intelligence | |
| 650 | 4 | |a Datenverarbeitung | |
| 650 | 4 | |a Informatik | |
| 650 | 4 | |a Ingenieurwissenschaften | |
| 700 | 1 | |a Neittaanmäki, Pekka |e Sonstige |4 oth | |
| 700 | 1 | |a Repin, Sergey |e Sonstige |4 oth | |
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| adam_text | ACCURACY VERIFICATION METHODS
/ MALI, OLLI
: 2014
TABLE OF CONTENTS / INHALTSVERZEICHNIS
1 ERRORS ARISING IN COMPUTER SIMULATION METHODS
1.1 GENERAL SCHEME
1.2 ERRORS OF MATHEMATICAL MODELS
1.3 APPROXIMATION ERRORS
1.4 NUMERICAL ERRORS
2 ERROR INDICATORS
2.1 ERROR INDICATORS AND ADAPTIVE NUMERICAL METHODS
2.1.1 ERROR INDICATORS FOR FEM SOLUTIONS
2.1.2 ACCURACY OF ERROR INDICATORS
2.2 ERROR INDICATORS FOR THE ENERGY NORM
2.2.1 ERROR INDICATORS BASED ON INTERPOLATION ESTIMATES
2.2.2 ERROR INDICATORS BASED ON APPROXIMATION OF THE ERROR FUNCTIONAL
2.2.3 ERROR INDICATORS OF THE RUNGE TYPE
2.3 ERROR INDICATORS FOR GOAL-ORIENTED QUANTITIES
2.3.1 ERROR INDICATORS RELYING ON THE SUPERCONVERGENCE OF AVERAGED
FLUXES IN THE PRIMAL AND ADJOINT PROBLEMS
2.3.2 ERROR INDICATORS USING THE SUPERCONVERGENCE OF APPROXIMATIONS IN
THE PRIMAL PROBLEM
2.3.3 ERROR INDICATORS BASED ON PARTIAL EQUILIBRATION OF FLUXES IN THE
ORIGINAL PROBLEM
3 GUARANTEED ERROR BOUNDS I
3.1 ORDINARY DIFFERENTIAL EQUATIONS
3.1.1 DERIVATION OF GUARANTEED ERROR BOUNDS
3.1.2 COMPUTATION OF ERROR BOUNDS
3.2 PARTIAL DIFFERENTIAL EQUATIONS
3.2.1 MAXIMAL DEVIATION FROM THE EXACT SOLUTION
3.2.2 MINIMAL DEVIATION FROM THE EXACT SOLUTION
3.2.3 PARTICULAR CASES
3.2.4 PROBLEMS WITH MIXED BOUNDARY CONDITIONS
3.2.5 ESTIMATES OF GLOBAL CONSTANTS ENTERING THE MAJORANT
3.2.6 ERROR MAJORANTS BASED ON POINCAR´E INEQUALITIES
3.2.7 ESTIMATES WITH PARTIALLY EQUILIBRATED FLUXES
3.3 ERROR CONTROL ALGORITHMS
3.3.1 GLOBAL MINIMIZATION OF THE MAJORANT
3.3.2 GETTING AN ERROR BOUND BY LOCAL PROCEDURES
3.4 INDICATORS BASED ON ERROR MAJORANTS
3.5 APPLICATIONS TO ADAPTIVE METHODS
3.6 COMBINED (PRIMAL-DUAL) ERROR NORMS AND THE MAJORANT
4 GUARANTEED ERROR BOUNDS II
4.1 LINEAR ELASTICITY
4.1.1 INTRODUCTION
4.1.2 EULER–BERNOULLI BEAM
4.1.3 THE KIRCHHOFF–LOVE ARCH MODEL
4.1.4 THE KIRCHHOFF–LOVE PLATE
4.1.5 THE REISSNER–MINDLIN PLATE
4.1.6 3D LINEAR ELASTICITY
4.1.7 THE PLANE STRESS MODEL
4.1.8 THE PLANE STRAIN MODEL
4.2 THE STOKES PROBLEM
4.2.1 DIVERGENCE-FREE APPROXIMATIONS
4.2.2 APPROXIMATIONS WITH NONZERO DIVERGENCE
4.2.3 STOKES PROBLEM IN ROTATING SYSTEM
4.3 A SIMPLE MAXWELL TYPE PROBLEM
4.3.1 ESTIMATES OF DEVIATIONS FROM EXACT SOLUTIONS
4.3.2 NUMERICAL EXAMPLES
4.4 GENERALIZATIONS
4.4.1 ERROR MAJORANT
4.4.2 ERROR MINORANT
5 ERRORS GENERATED BY UNCERTAIN DATA
5.1 MATHEMATICAL MODELS WITH INCOMPLETELY KNOWN DATA
5.2 THE ACCURACY LIMIT
5.3 ESTIMATES OF THE WORST AND BEST CASE SCENARIO ERRORS
5.4 TWO-SIDED BOUNDS OF THE RADIUS OF THE SOLUTION SET
5.5 COMPUTABLE ESTIMATES OF THE RADIUS OF THE SOLUTION SET
5.5.1 USING THE MAJORANT
5.5.2 USING A REFERENCE SOLUTION
5.5.3 AN ADVANCED LOWER BOUND
5.6 MULTIPLE SOURCES OF INDETERMINACY
5.6.1 INCOMPLETELY KNOWN RIGHT-HAND SIDE
5.6.2 THE REACTION DIFFUSION PROBLEM
5.7 ERROR INDICATION AND INDETERMINATE DATA
5.8 LINEAR ELASTICITY WITH INCOMPLETELY KNOWN POISSON RATIO
5.8.1 SENSITIVITY OF THE ENERGY FUNCTIONAL
5.8.2 EXAMPLE: AXISYMMETRIC MODEL
6 OVERVIEW OF OTHER RESULTS AND OPEN PROBLEMS
6.1 ERROR ESTIMATES FOR APPROXIMATIONS VIOLATING CONFORMITY
6.2 LINEAR ELLIPTIC EQUATIONS
6.3 TIME-DEPENDENT PROBLEMS
6.4 OPTIMAL CONTROL AND INVERSE PROBLEMS
6.5 NONLINEAR BOUNDARY VALUE PROBLEMS
6.5.1 VARIATIONAL INEQUALITIES
6.5.2 ELASTOPLASTICITY
6.5.3 PROBLEMS WITH POWER GROWTH ENERGY FUNCTIONALS
6.6 MODELING ERRORS
6.7 ERROR BOUNDS FOR ITERATION METHODS
6.7.1 GENERAL ITERATION ALGORITHM
6.7.2 A PRIORI ESTIMATES OF ERRORS
6.7.3 A POSTERIORI ESTIMATES OF ERRORS
6.7.4 ADVANCED FORMS OF ERROR BOUNDS
6.7.5 SYSTEMS OF LINEAR SIMULTANEOUS EQUATIONS
6.7.6 ORDINARY DIFFERENTIAL EQUATIONS
6.8 ROUNDOFF ERRORS
6.9 OPEN PROBLEMS
A MATHEMATICAL BACKGROUND
A.1 VECTORS AND TENSORS
A.2 SPACES OF FUNCTIONS
A.2.1 LEBESGUE AND SOBOLEV SPACES
A.2.2 BOUNDARY TRACES
A.2.3 LINEAR FUNCTIONALS
A.3 INEQUALITIES
A.3.1 THE HOELDER INEQUALITY
A.3.2 THE POINCARE AND FRIEDRICHS INEQUALITIES
A.3.3 KORN’S INEQUALITY
A.3.4 LBB INEQUALITY
A.4 CONVEX FUNCTIONALS
B BOUNDARY VALUE PROBLEMS
B.1 GENERALIZED SOLUTIONS OF BOUNDARY VALUE PROBLEMS
B.2 VARIATIONAL STATEMENTS OF ELLIPTIC BOUNDARY VALUE PROBLEMS
B.3 SADDLE POINT STATEMENTS OF ELLIPTIC BOUNDARY VALUE PROBLEMS
B.3.1 INTRODUCTION TO THE THEORY OF SADDLE POINTS
B.3.2 SADDLE POINT STATEMENTS OF LINEAR ELLIPTIC PROBLEMS
B.3.3 SADDLE POINT STATEMENTS OF NONLINEAR VARIATIONAL PROBLEMS
B.4 NUMERICAL METHODS
B.4.1 FINITE DIFFERENCE METHODS
B.4.2 VARIATIONAL DIFFERENCE METHODS
B.4.3 PETROV–GALERKIN METHODS
B.4.4 MIXED FINITE ELEMENT METHODS
B.4.5 TREFFTZ METHODS
B.4.6 FINITE VOLUME METHODS
B.4.7 DISCONTINUOUS GALERKIN METHODS
B.4.8 FICTITIOUS DOMAIN METHODS
C A PRIORI VERIFICATION OF ACCURACY
C.1 PROJECTION ERROR ESTIMATE
C.2 INTERPOLATION THEORY IN SOBOLEV SPACES
C.3 A PRIORI CONVERGENCE RATE ESTIMATES
C.4 A PRIORI ERROR ESTIMATES FOR MIXED FEM
REFERENCES
NOTATION
INDEX
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
ACCURACY VERIFICATION METHODS
/ MALI, OLLI
: 2014
ABSTRACT / INHALTSTEXT
THE IMPORTANCE OF ACCURACY VERIFICATION METHODS WAS UNDERSTOOD AT THE
VERY BEGINNING OF THE DEVELOPMENT OF NUMERICAL ANALYSIS. RECENT DECADES
HAVE SEEN A RAPID GROWTH OF RESULTS RELATED TO ADAPTIVE NUMERICAL
METHODS AND A POSTERIORI ESTIMATES. HOWEVER, IN THIS IMPORTANT AREA
THERE OFTEN EXISTS A NOTICEABLE GAP BETWEEN MATHEMATICIANS CREATING THE
THEORY AND RESEARCHERS DEVELOPING APPLIED ALGORITHMS THAT COULD BE USED
IN ENGINEERING AND SCIENTIFIC COMPUTATIONS FOR GUARANTEED AND EFFICIENT
ERROR CONTROL. THE GOALS OF THE BOOK ARE TO (1) GIVE A TRANSPARENT
EXPLANATION OF THE UNDERLYING MATHEMATICAL THEORY IN A STYLE ACCESSIBLE
NOT ONLY TO ADVANCED NUMERICAL ANALYSTS BUT ALSO TO ENGINEERS AND
STUDENTS; (2) PRESENT DETAILED STEP-BY-STEP ALGORITHMS THAT FOLLOW FROM
A THEORY; (3) DISCUSS THEIR ADVANTAGES AND DRAWBACKS, AREAS OF
APPLICABILITY, GIVE RECOMMENDATIONS AND EXAMPLES
DIESES SCHRIFTSTUECK WURDE MASCHINELL ERZEUGT.
|
| any_adam_object | 1 |
| author | Mali, Olli |
| author_facet | Mali, Olli |
| author_role | aut |
| author_sort | Mali, Olli |
| author_variant | o m om |
| building | Verbundindex |
| bvnumber | BV041471113 |
| collection | ZDB-2-ENG |
| contents | 1 Errors Arising In Computer Simulation Methods -- 1.1 General scheme -- 1.2 Errors of mathematical models -- 1.3 Approximation errors -- 1.4 Numerical errors -- 2 Error Indicators -- 2.1 Error indicators and adaptive numerical methods -- 2.1.1 Error indicators for FEM solutions -- 2.1.2 Accuracy of error indicators -- 2.2 Error indicators for the energy norm -- 2.2.1 Error indicators based on interpolation estimates -- 2.2.2 Error indicators based on approximation of the error functional -- 2.2.3 Error indicators of the Runge type -- 2.3 Error indicators for goal-oriented quantities -- 2.3.1 Error indicators relying on the superconvergence of averaged fluxes in the primal and adjoint problems -- 2.3.2 Error indicators using the superconvergence of approximations in the primal problem -- 2.3.3 Error indicators based on partial equilibration of fluxes in the original problem -- 3 Guaranteed Error Bounds I -- 3.1 Ordinary differential equations -- 3.1.1 Derivation of guaranteed error bounds -- 3.1.2 Computation of error bounds -- 3.2 Partial differential equations -- 3.2.1 Maximal deviation from the exact solution -- 3.2.2 Minimal deviation from the exact solution -- 3.2.3 Particular cases -- 3.2.4 Problems with mixed boundary conditions -- 3.2.5 Estimates of global constants entering the majorant -- 3.2.6 Error majorants based on Poincar'e inequalities -- 3.2.7 Estimates with partially equilibrated fluxes -- 3.3 Error control algorithms -- 3.3.1 Global minimization of the majorant -- 3.3.2 Getting an error bound by local procedures -- 3.4 Indicators based on error majorants -- 3.5 Applications to adaptive methods -- 3.6 Combined (primal-dual) error norms and the majorant -- 4 Guaranteed Error Bounds II -- 4.1 Linear elasticity -- 4.1.1 Introduction -- 4.1.2 Euler–Bernoulli beam -- 4.1.3 The Kirchhoff–Love arch model -- 4.1.4 The Kirchhoff–Love plate -- 4.1.5 The Reissner–Mindlin plate -- 4.1.6 3D linear elasticity -- 4.1.7 The plane stress model -- 4.1.8 The plane strain model -- 4.2 The Stokes Problem -- 4.2.1 Divergence-free approximations -- 4.2.2 Approximations with nonzero divergence -- 4.2.3 Stokes problem in rotating system -- 4.3 A simple Maxwell type problem -- 4.3.1 Estimates of deviations from exact solutions -- 4.3.2 Numerical examples -- 4.4 Generalizations -- 4.4.1 Error majorant -- 4.4.2 Error minorant -- 5 Errors Generated By Uncertain Data -- 5.1 Mathematical models with incompletely known data -- 5.2 The accuracy limit -- 5.3 Estimates of the worst and best case scenario errors -- 5.4 Two-sided bounds of the radius of the solution set -- 5.5 Computable estimates of the radius of the solution set -- 5.5.1 Using the majorant -- 5.5.2 Using a reference solution -- 5.5.3 An advanced lower bound -- 5.6 Multiple sources of indeterminacy -- 5.6.1 Incompletely known right-hand side -- 5.6.2 The reaction diffusion problem -- 5.7 Error indication and indeterminate data -- 5.8 Linear elasticity with incompletely known Poisson ratio -- 5.8.1 Sensitivity of the energy functional -- 5.8.2 Example: axisymmetric model -- 6 Overview Of Other Results And Open Problems -- 6.1 Error estimates for approximations violating conformity -- 6.2 Linear elliptic equations -- 6.3 Time-dependent problems -- 6.4 Optimal control and inverse problems -- 6.5 Nonlinear boundary value problems -- 6.5.1 Variational inequalities -- 6.5.2 Elastoplasticity -- 6.5.3 Problems with power growth energy functionals -- 6.6 Modeling errors -- 6.7 Error bounds for iteration methods -- 6.7.1 General iteration algorithm -- 6.7.2 A priori estimates of errors -- 6.7.3 A posteriori estimates of errors -- 6.7.4 Advanced forms of error bounds -- 6.7.5 Systems of linear simultaneous equations -- 6.7.6 Ordinary differential equations -- 6.8 Roundoff errors -- 6.9 Open problems -- A Mathematical Background -- A.1 Vectors and tensors -- A.2 Spaces of functions -- A.2.1 Lebesgue and Sobolev spaces -- A.2.2 Boundary traces -- A.2.3 Linear functionals -- A.3 Inequalities -- A.3.1 The Hölder inequality -- A.3.2 The Poincaré and Friedrichs inequalities -- A.3.3 Korn’s inequality -- A.3.4 LBB inequality -- A.4 Convex functionals -- B Boundary Value Problems -- B.1 Generalized solutions of boundary value problems -- B.2 Variational statements of elliptic boundary value problems -- B.3 Saddle point statements of elliptic boundary value problems -- B.3.1 Introduction to the theory of saddle points -- B.3.2 Saddle point statements of linear elliptic problems -- B.3.3 Saddle point statements of nonlinear variational problems -- B.4 Numerical methods -- B.4.1 Finite difference methods -- B.4.2 Variational difference methods -- B.4.3 Petrov–Galerkin methods -- B.4.4 Mixed finite element methods -- B.4.5 Trefftz methods -- B.4.6 Finite volume methods -- B.4.7 Discontinuous Galerkin methods -- B.4.8 Fictitious domain methods -- C A Priori Verification Of Accuracy -- C.1 Projection error estimate -- C.2 Interpolation theory in Sobolev spaces -- C.3 A priori convergence rate estimates -- C.4 A priori error estimates for mixed FEM -- References -- Notation -- Index |
| ctrlnum | (OCoLC)1195528078 (DE-599)BVBBV041471113 |
| dewey-full | 518 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 518 - Numerical analysis |
| dewey-raw | 518 |
| dewey-search | 518 |
| dewey-sort | 3518 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-94-007-7581-7 |
| format | Electronic eBook |
| fullrecord | <?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>09214nmm a2200721zcb4500</leader><controlfield tag="001">BV041471113</controlfield><controlfield tag="003">DE-604</controlfield><controlfield tag="005">00000000000000.0</controlfield><controlfield tag="007">cr|uuu---uuuuu</controlfield><controlfield tag="008">131210s2014 |||| o||u| ||||||eng d</controlfield><datafield tag="020" ind1=" " ind2=" "><subfield code="a">9789400775817</subfield><subfield code="9">978-94-007-7581-7</subfield></datafield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/978-94-007-7581-7</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(OCoLC)1195528078</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-599)BVBBV041471113</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-604</subfield><subfield code="b">ger</subfield><subfield code="e">aacr</subfield></datafield><datafield tag="041" ind1="0" ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="049" ind1=" " ind2=" "><subfield code="a">DE-Aug4</subfield><subfield code="a">DE-92</subfield><subfield code="a">DE-634</subfield><subfield code="a">DE-859</subfield><subfield code="a">DE-898</subfield><subfield code="a">DE-573</subfield><subfield code="a">DE-861</subfield><subfield code="a">DE-706</subfield><subfield code="a">DE-863</subfield><subfield code="a">DE-862</subfield></datafield><datafield tag="082" ind1="0" ind2=" "><subfield code="a">518</subfield><subfield code="2">23</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Mali, Olli</subfield><subfield code="e">Verfasser</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Accuracy Verification Methods</subfield><subfield code="b">Theory and Algorithms</subfield><subfield code="c">by Olli Mali, Pekka Neittaanmäki, Sergey Repin</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2014</subfield></datafield><datafield tag="300" ind1=" " ind2=" "><subfield code="a">1 Online-Ressource (XIII, 355 p.)</subfield><subfield code="b">75 illus</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="b">cr</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="490" ind1="0" ind2=" "><subfield code="a">Computational Methods in Applied Sciences</subfield><subfield code="v">32</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">The importance of accuracy verification methods was understood at the very beginning of the development of numerical analysis. Recent decades have seen a rapid growth of results related to adaptive numerical methods and a posteriori estimates. However, in this important area there often exists a noticeable gap between mathematicians creating the theory and researchers developing applied algorithms that could be used in engineering and scientific computations for guaranteed and efficient error control. The goals of the book are to (1) give a transparent explanation of the underlying mathematical theory in a style accessible not only to advanced numerical analysts but also to engineers and students; (2) present detailed step-by-step algorithms that follow from a theory; (3) discuss their advantages and drawbacks, areas of applicability, give recommendations and examples</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">1 Errors Arising In Computer Simulation Methods -- 1.1 General scheme -- 1.2 Errors of mathematical models -- 1.3 Approximation errors -- 1.4 Numerical errors -- 2 Error Indicators -- 2.1 Error indicators and adaptive numerical methods -- 2.1.1 Error indicators for FEM solutions -- 2.1.2 Accuracy of error indicators -- 2.2 Error indicators for the energy norm -- 2.2.1 Error indicators based on interpolation estimates -- 2.2.2 Error indicators based on approximation of the error functional -- 2.2.3 Error indicators of the Runge type -- 2.3 Error indicators for goal-oriented quantities -- 2.3.1 Error indicators relying on the superconvergence of averaged fluxes in the primal and adjoint problems -- 2.3.2 Error indicators using the superconvergence of approximations in the primal problem -- 2.3.3 Error indicators based on partial equilibration of fluxes in the original problem -- 3 Guaranteed Error Bounds I -- 3.1 Ordinary differential equations --</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">3.1.1 Derivation of guaranteed error bounds -- 3.1.2 Computation of error bounds -- 3.2 Partial differential equations -- 3.2.1 Maximal deviation from the exact solution -- 3.2.2 Minimal deviation from the exact solution -- 3.2.3 Particular cases -- 3.2.4 Problems with mixed boundary conditions -- 3.2.5 Estimates of global constants entering the majorant -- 3.2.6 Error majorants based on Poincar'e inequalities -- 3.2.7 Estimates with partially equilibrated fluxes -- 3.3 Error control algorithms -- 3.3.1 Global minimization of the majorant -- 3.3.2 Getting an error bound by local procedures -- 3.4 Indicators based on error majorants -- 3.5 Applications to adaptive methods -- 3.6 Combined (primal-dual) error norms and the majorant -- 4 Guaranteed Error Bounds II -- 4.1 Linear elasticity -- 4.1.1 Introduction -- 4.1.2 Euler–Bernoulli beam -- 4.1.3 The Kirchhoff–Love arch model -- 4.1.4 The Kirchhoff–Love plate -- 4.1.5 The Reissner–Mindlin plate -- 4.1.6 3D linear elasticity --</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">4.1.7 The plane stress model -- 4.1.8 The plane strain model -- 4.2 The Stokes Problem -- 4.2.1 Divergence-free approximations -- 4.2.2 Approximations with nonzero divergence -- 4.2.3 Stokes problem in rotating system -- 4.3 A simple Maxwell type problem -- 4.3.1 Estimates of deviations from exact solutions -- 4.3.2 Numerical examples -- 4.4 Generalizations -- 4.4.1 Error majorant -- 4.4.2 Error minorant -- 5 Errors Generated By Uncertain Data -- 5.1 Mathematical models with incompletely known data -- 5.2 The accuracy limit -- 5.3 Estimates of the worst and best case scenario errors -- 5.4 Two-sided bounds of the radius of the solution set -- 5.5 Computable estimates of the radius of the solution set -- 5.5.1 Using the majorant -- 5.5.2 Using a reference solution -- 5.5.3 An advanced lower bound -- 5.6 Multiple sources of indeterminacy -- 5.6.1 Incompletely known right-hand side -- 5.6.2 The reaction diffusion problem -- 5.7 Error indication and indeterminate data --</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">5.8 Linear elasticity with incompletely known Poisson ratio -- 5.8.1 Sensitivity of the energy functional -- 5.8.2 Example: axisymmetric model -- 6 Overview Of Other Results And Open Problems -- 6.1 Error estimates for approximations violating conformity -- 6.2 Linear elliptic equations -- 6.3 Time-dependent problems -- 6.4 Optimal control and inverse problems -- 6.5 Nonlinear boundary value problems -- 6.5.1 Variational inequalities -- 6.5.2 Elastoplasticity -- 6.5.3 Problems with power growth energy functionals -- 6.6 Modeling errors -- 6.7 Error bounds for iteration methods -- 6.7.1 General iteration algorithm -- 6.7.2 A priori estimates of errors -- 6.7.3 A posteriori estimates of errors -- 6.7.4 Advanced forms of error bounds -- 6.7.5 Systems of linear simultaneous equations -- 6.7.6 Ordinary differential equations -- 6.8 Roundoff errors -- 6.9 Open problems -- A Mathematical Background -- A.1 Vectors and tensors -- A.2 Spaces of functions -- A.2.1 Lebesgue and Sobolev spaces --</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">A.2.2 Boundary traces -- A.2.3 Linear functionals -- A.3 Inequalities -- A.3.1 The Hölder inequality -- A.3.2 The Poincaré and Friedrichs inequalities -- A.3.3 Korn’s inequality -- A.3.4 LBB inequality -- A.4 Convex functionals -- B Boundary Value Problems -- B.1 Generalized solutions of boundary value problems -- B.2 Variational statements of elliptic boundary value problems -- B.3 Saddle point statements of elliptic boundary value problems -- B.3.1 Introduction to the theory of saddle points -- B.3.2 Saddle point statements of linear elliptic problems -- B.3.3 Saddle point statements of nonlinear variational problems -- B.4 Numerical methods -- B.4.1 Finite difference methods -- B.4.2 Variational difference methods -- B.4.3 Petrov–Galerkin methods -- B.4.4 Mixed finite element methods -- B.4.5 Trefftz methods -- B.4.6 Finite volume methods -- B.4.7 Discontinuous Galerkin methods -- B.4.8 Fictitious domain methods -- C A Priori Verification Of Accuracy --</subfield></datafield><datafield tag="505" ind1="0" ind2=" "><subfield code="a">C.1 Projection error estimate -- C.2 Interpolation theory in Sobolev spaces -- C.3 A priori convergence rate estimates -- C.4 A priori error estimates for mixed FEM -- References -- Notation -- Index</subfield></datafield><datafield 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| id | DE-604.BV041471113 |
| illustrated | Not Illustrated |
| indexdate | 2025-02-20T06:39:01Z |
| institution | BVB |
| isbn | 9789400775817 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-026917255 |
| oclc_num | 1195528078 |
| open_access_boolean | |
| owner | DE-Aug4 DE-92 DE-634 DE-859 DE-898 DE-BY-UBR DE-573 DE-861 DE-706 DE-863 DE-BY-FWS DE-862 DE-BY-FWS |
| owner_facet | DE-Aug4 DE-92 DE-634 DE-859 DE-898 DE-BY-UBR DE-573 DE-861 DE-706 DE-863 DE-BY-FWS DE-862 DE-BY-FWS |
| physical | 1 Online-Ressource (XIII, 355 p.) 75 illus |
| psigel | ZDB-2-ENG |
| publishDate | 2014 |
| publishDateSearch | 2014 |
| publishDateSort | 2014 |
| record_format | marc |
| series2 | Computational Methods in Applied Sciences |
| spellingShingle | Mali, Olli Accuracy Verification Methods Theory and Algorithms 1 Errors Arising In Computer Simulation Methods -- 1.1 General scheme -- 1.2 Errors of mathematical models -- 1.3 Approximation errors -- 1.4 Numerical errors -- 2 Error Indicators -- 2.1 Error indicators and adaptive numerical methods -- 2.1.1 Error indicators for FEM solutions -- 2.1.2 Accuracy of error indicators -- 2.2 Error indicators for the energy norm -- 2.2.1 Error indicators based on interpolation estimates -- 2.2.2 Error indicators based on approximation of the error functional -- 2.2.3 Error indicators of the Runge type -- 2.3 Error indicators for goal-oriented quantities -- 2.3.1 Error indicators relying on the superconvergence of averaged fluxes in the primal and adjoint problems -- 2.3.2 Error indicators using the superconvergence of approximations in the primal problem -- 2.3.3 Error indicators based on partial equilibration of fluxes in the original problem -- 3 Guaranteed Error Bounds I -- 3.1 Ordinary differential equations -- 3.1.1 Derivation of guaranteed error bounds -- 3.1.2 Computation of error bounds -- 3.2 Partial differential equations -- 3.2.1 Maximal deviation from the exact solution -- 3.2.2 Minimal deviation from the exact solution -- 3.2.3 Particular cases -- 3.2.4 Problems with mixed boundary conditions -- 3.2.5 Estimates of global constants entering the majorant -- 3.2.6 Error majorants based on Poincar'e inequalities -- 3.2.7 Estimates with partially equilibrated fluxes -- 3.3 Error control algorithms -- 3.3.1 Global minimization of the majorant -- 3.3.2 Getting an error bound by local procedures -- 3.4 Indicators based on error majorants -- 3.5 Applications to adaptive methods -- 3.6 Combined (primal-dual) error norms and the majorant -- 4 Guaranteed Error Bounds II -- 4.1 Linear elasticity -- 4.1.1 Introduction -- 4.1.2 Euler–Bernoulli beam -- 4.1.3 The Kirchhoff–Love arch model -- 4.1.4 The Kirchhoff–Love plate -- 4.1.5 The Reissner–Mindlin plate -- 4.1.6 3D linear elasticity -- 4.1.7 The plane stress model -- 4.1.8 The plane strain model -- 4.2 The Stokes Problem -- 4.2.1 Divergence-free approximations -- 4.2.2 Approximations with nonzero divergence -- 4.2.3 Stokes problem in rotating system -- 4.3 A simple Maxwell type problem -- 4.3.1 Estimates of deviations from exact solutions -- 4.3.2 Numerical examples -- 4.4 Generalizations -- 4.4.1 Error majorant -- 4.4.2 Error minorant -- 5 Errors Generated By Uncertain Data -- 5.1 Mathematical models with incompletely known data -- 5.2 The accuracy limit -- 5.3 Estimates of the worst and best case scenario errors -- 5.4 Two-sided bounds of the radius of the solution set -- 5.5 Computable estimates of the radius of the solution set -- 5.5.1 Using the majorant -- 5.5.2 Using a reference solution -- 5.5.3 An advanced lower bound -- 5.6 Multiple sources of indeterminacy -- 5.6.1 Incompletely known right-hand side -- 5.6.2 The reaction diffusion problem -- 5.7 Error indication and indeterminate data -- 5.8 Linear elasticity with incompletely known Poisson ratio -- 5.8.1 Sensitivity of the energy functional -- 5.8.2 Example: axisymmetric model -- 6 Overview Of Other Results And Open Problems -- 6.1 Error estimates for approximations violating conformity -- 6.2 Linear elliptic equations -- 6.3 Time-dependent problems -- 6.4 Optimal control and inverse problems -- 6.5 Nonlinear boundary value problems -- 6.5.1 Variational inequalities -- 6.5.2 Elastoplasticity -- 6.5.3 Problems with power growth energy functionals -- 6.6 Modeling errors -- 6.7 Error bounds for iteration methods -- 6.7.1 General iteration algorithm -- 6.7.2 A priori estimates of errors -- 6.7.3 A posteriori estimates of errors -- 6.7.4 Advanced forms of error bounds -- 6.7.5 Systems of linear simultaneous equations -- 6.7.6 Ordinary differential equations -- 6.8 Roundoff errors -- 6.9 Open problems -- A Mathematical Background -- A.1 Vectors and tensors -- A.2 Spaces of functions -- A.2.1 Lebesgue and Sobolev spaces -- A.2.2 Boundary traces -- A.2.3 Linear functionals -- A.3 Inequalities -- A.3.1 The Hölder inequality -- A.3.2 The Poincaré and Friedrichs inequalities -- A.3.3 Korn’s inequality -- A.3.4 LBB inequality -- A.4 Convex functionals -- B Boundary Value Problems -- B.1 Generalized solutions of boundary value problems -- B.2 Variational statements of elliptic boundary value problems -- B.3 Saddle point statements of elliptic boundary value problems -- B.3.1 Introduction to the theory of saddle points -- B.3.2 Saddle point statements of linear elliptic problems -- B.3.3 Saddle point statements of nonlinear variational problems -- B.4 Numerical methods -- B.4.1 Finite difference methods -- B.4.2 Variational difference methods -- B.4.3 Petrov–Galerkin methods -- B.4.4 Mixed finite element methods -- B.4.5 Trefftz methods -- B.4.6 Finite volume methods -- B.4.7 Discontinuous Galerkin methods -- B.4.8 Fictitious domain methods -- C A Priori Verification Of Accuracy -- C.1 Projection error estimate -- C.2 Interpolation theory in Sobolev spaces -- C.3 A priori convergence rate estimates -- C.4 A priori error estimates for mixed FEM -- References -- Notation -- Index Computer science Electronic data processing Numerical analysis Engineering Computer Science Numeric Computing Computational Science and Engineering Numerical Analysis Numerical and Computational Physics Computational Intelligence Datenverarbeitung Informatik Ingenieurwissenschaften |
| title | Accuracy Verification Methods Theory and Algorithms |
| title_auth | Accuracy Verification Methods Theory and Algorithms |
| title_exact_search | Accuracy Verification Methods Theory and Algorithms |
| title_full | Accuracy Verification Methods Theory and Algorithms by Olli Mali, Pekka Neittaanmäki, Sergey Repin |
| title_fullStr | Accuracy Verification Methods Theory and Algorithms by Olli Mali, Pekka Neittaanmäki, Sergey Repin |
| title_full_unstemmed | Accuracy Verification Methods Theory and Algorithms by Olli Mali, Pekka Neittaanmäki, Sergey Repin |
| title_short | Accuracy Verification Methods |
| title_sort | accuracy verification methods theory and algorithms |
| title_sub | Theory and Algorithms |
| topic | Computer science Electronic data processing Numerical analysis Engineering Computer Science Numeric Computing Computational Science and Engineering Numerical Analysis Numerical and Computational Physics Computational Intelligence Datenverarbeitung Informatik Ingenieurwissenschaften |
| topic_facet | Computer science Electronic data processing Numerical analysis Engineering Computer Science Numeric Computing Computational Science and Engineering Numerical Analysis Numerical and Computational Physics Computational Intelligence Datenverarbeitung Informatik Ingenieurwissenschaften |
| url | https://doi.org/10.1007/978-94-007-7581-7 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026917255&sequence=000001&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=026917255&sequence=000003&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
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