Accuracy Verification Methods Theory and Algorithms /
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 not...
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Autor principal: | |
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Otros Autores: | , |
Formato: | Libro electrónico |
Lenguaje: | Inglés |
Publicado: |
Dordrecht :
Springer Netherlands : Imprint: Springer,
2014.
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Colección: | Computational Methods in Applied Sciences,
32 |
Materias: | |
Acceso en línea: | http://dx.doi.org/10.1007/978-94-007-7581-7 |
Aporte de: | Registro referencial: Solicitar el recurso aquí |
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024 | 7 | |a 10.1007/978-94-007-7581-7 |2 doi | |
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072 | 7 | |a UYA |2 bicssc | |
072 | 7 | |a COM051300 |2 bisacsh | |
100 | 1 | |a Mali, Olli. |9 262208 | |
245 | 1 | 0 | |a Accuracy Verification Methods |h [libro electrónico] : ; |b Theory and Algorithms / |c by Olli Mali, Pekka Neittaanmaki, Sergey Repin. |
260 | 1 | |a Dordrecht : |b Springer Netherlands : |b Imprint: Springer, |c 2014. | |
300 | |a xiii, 355 p. : |b il. | ||
336 | |a text |b txt |2 rdacontent | ||
337 | |a computer |b c |2 rdamedia | ||
338 | |a online resource |b cr |2 rdacarrier | ||
347 | |a text file |b PDF |2 rda | ||
490 | 1 | |a Computational Methods in Applied Sciences, |x 1871-3033 ; |v 32 | |
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 -- 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 Eulerâ_"Bernoulli beam -- 4.1.3 The Kirchhoffâ_"Love arch model -- 4.1.4 The Kirchhoffâ_"Love plate -- 4.1.5 The Reissnerâ_"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 Kornâ_Ts 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 Petrovâ_"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. | |
520 | |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. Â 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. | ||
650 | 0 | |a Computer mathematics. |9 259612 | |
650 | 0 | |a Physics. |9 259968 | |
650 | 0 | |a Computational intelligence. |9 259845 | |
650 | 1 | 4 | |a Computer Science. |9 260143 |
650 | 2 | 4 | |a Numeric Computing. |9 260925 |
650 | 2 | 4 | |a Computational Science and Engineering. |9 260310 |
650 | 2 | 4 | |a Numerical Analysis. |9 260924 |
650 | 2 | 4 | |a Numerical and Computational Physics. |9 260529 |
700 | 1 | |a Neittaanmaki, Pekka, |9 260531 | |
700 | 1 | |a Repin, Sergey. |9 262209 | |
776 | 0 | 8 | |i Printed edition: |z 9789400775800 |
856 | 4 | 0 | |u http://dx.doi.org/10.1007/978-94-007-7581-7 |
912 | |a ZDB-2-ENG | ||
929 | |a COM | ||
942 | |c EBK |6 _ | ||
950 | |a Engineering (Springer-11647) | ||
999 | |a SKV |c 28312 |d 28312 |