Accuracy Verification Methods: Theory and Algorithms

Author:   Olli Mali ,  Pekka Neittaanmäki ,  Sergey Repin
Publisher:   Springer
Edition:   Softcover reprint of the original 1st ed. 2014
Volume:   32
ISBN:  

9789402404982


Pages:   355
Publication Date:   17 September 2016
Format:   Paperback
Availability:   In Print   Availability explained
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Our Price $290.37 Quantity:  
Add to Cart

Share |

Accuracy Verification Methods: Theory and Algorithms


Add your own review!

Overview

Full Product Details

Author:   Olli Mali ,  Pekka Neittaanmäki ,  Sergey Repin
Publisher:   Springer
Imprint:   Springer
Edition:   Softcover reprint of the original 1st ed. 2014
Volume:   32
Dimensions:   Width: 15.50cm , Height: 2.00cm , Length: 23.50cm
Weight:   5.621kg
ISBN:  

9789402404982


ISBN 10:   9402404988
Pages:   355
Publication Date:   17 September 2016
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Paperback
Publisher's Status:   Active
Availability:   In Print   Availability explained
This item will be ordered in for you from one of our suppliers. Upon receipt, we will promptly dispatch it out to you. For in store availability, please contact us.

Table of 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.

Reviews

From the book reviews: An up-to-date monograph for researchers, practitioners and advanced students in scientific computing. It tries to bridge the gap between theory and practice in error estimation spanning the whole modeling and simulation process. (A. Bultheel, Mathematical Reviews, July, 2014)


From the book reviews: “An up-to-date monograph for researchers, practitioners and advanced students in scientific computing. It tries to bridge the gap between theory and practice in error estimation spanning the whole modeling and simulation process.” (A. Bultheel, Mathematical Reviews, July, 2014)


From the book reviews: An up-to-date monograph for researchers, practitioners and advanced students in scientific computing. It tries to bridge the gap between theory and practice in error estimation spanning the whole modeling and simulation process. (A. Bultheel, Mathematical Reviews, July, 2014)


From the book reviews: An up-to-date monograph for researchers, practitioners and advanced students in scientific computing. It tries to bridge the gap between theory and practice in error estimation spanning the whole modeling and simulation process. (A. Bultheel, Mathematical Reviews, July, 2014)


Author Information

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions
Latest Reading Guide

Aorrng

Shopping Cart
Your cart is empty
Shopping cart
Mailing List