Accuracy of Mathematical Models: Dimension Reduction, Homogenization, and Simplification

Description

Preface Introduction Basic notation Domains and operators Spaces of functions Convex functionals Functional inequalities Hölder type inequalities Friedrichs and Poincaré inequalities Inequalities for functions with zero mean traceson the boundary Korn's inequalities Inf–Sup condition Computable bounds of constantsin functional inequalities Constant in the Friedrichs inequality Constants in Poincaré-type inequalities Constants in trace-type inequalities Estimates of constants based on domain decomposition Distance to exact solutions A class of boundary value problems The main error identity Error measure Decomposition of the error measure Problems with linear F Error identities in vector form Difference between the exact solutions of two problems Linear problems Error relations in the general form Special case Primal-dual norms of errors in VY* Errors in the full primal-dual norm Majorant as a source of new models Non-homogeneous boundary conditions Applications to particular mathematical models Diffusion type models Mixed boundary conditions Problems with periodic boundary conditions Advanced estimates based on domain decomposition Elasticity Variational functionals with power growth Stokes problem Bingham problem Another error estimation method Validation of mathematical models Errors of numerical approximations Two-sided estimates of approximation errors Reduction of the set Q** Transformation of "426830A R(y*h),eh "526930B Using extra regularity of the exact solution Using an auxiliary finite-dimensional problem Applications to least squares type methods Nonconforming approximations Dimension reduction models Dimension reduction Second-order elliptic problems Basic problem Reduced problem Error generated by dimension reduction Particular cases Examples Dimension reduction in linear elasticity The plane stress problem The function Behavior of the modelling error as t0 Example Bending of elastic plates Statement of the problem The Kirchhoff–Love plate model Reconstruction of 3D displacements Reconstruction of 3D stresses Error estimates for plate-type domains Accuracy of the KL plate model Estimates of the modelling error Asymptotic behaviour of the error majorant Model simplification Model simplification based on the concept of energy Simplification of coefficients Second-order elliptic problems General elliptic problem Using extra regularity of Asymptotic rate of convergence of the error estimator E()in terms of the measure of the non-resolved geometry Geometrical simplification Simplification of the Dirichlet boundary Generalizations Simplification of the Neumann boundary Comments Problems with ``rough'' coefficients Modelling-discretization adaptation strategies Problems with uncertain data Elliptic homogenization Setting Mathematical homogenization via asymptotic expansions Properties of the homogenized problem Well-posedness of the homogenized equation Regularity estimates for the homogenized equation Regularity estimates for the cell problem Convergence of the first-order approximation Discretization Error estimation General comments Estimates of the modelling error Error of the fully discrete first-order approximation Comments Regularity and embedding constants Modeling-discretization strategies Multiscale problems Conversion of models Regularization of models Adding a regularizing term Smoothing Prox-type regularization Errors of penalty-type models General approach Variational problems defined in subspaces Fictitious domain methods Linearization Errors of time-incremental models bold0mu mumu W1,pW1,pdottedW1,pW1,pW1,pW1,p-regularity constant for second-order elliptic problemswith nonsmooth coefficients Bibliography List of Notations Index