Accuracy Verification Methods

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 TheStokes 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 Friedrichsinequalities.- 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.