A Posteriori Error Analysis Via Duality Theory: With Applications in Modeling and Numerical Approximations (Advances in Mechanics and Mathematics Book 8)

this Volume Provides A Posteriori Error Analysis For Mathematical Idealizations In Modeling Boundary Value Problems, Especially Those Arising In Mechanical Applications, And For Numerical Approximations Of Numerous Nonlinear Variational Problems. The Author Avoids Giving The Results In The Most General, Abstract Form So That It Is Easier For The Reader To Understand More Clearly The Essential Ideas Involved. Many Examples Are Included To Show The Usefulness Of The Derived Error Estimates. audience this Volume Is Suitable For Researchers And Graduate Students In Applied And Computational Mathematics, And In Engineering. Cover......Page 1 Advances in Mechanics and Mathematics Volume 8......Page 3 A Posteriori Error Analysis Via Duality Theory: With Applications in Modeling and Numerical Approximations......Page 4 0387235361......Page 5 Contents......Page 6 List of Figures......Page 8 List of Tables......Page 12 Preface......Page 16 1.1 Introduction......Page 18 1.2 Some basic notions from functional analysis......Page 22 1.3 Function spaces......Page 24 1.4 Weak formulation of boundary value problems......Page 33 1.5 Best constants in some Sobolev inequalities......Page 37 1.6 Singularities of elliptic problems on planar nonsmooth domains......Page 42 1.7 An introduction of elliptic variational inequalities......Page 46 1.8 Finite element method, error estimates......Page 53 2.1 Convex sets and convex functions......Page 64 2.2 Hahn-Banach theorem and separation of convex sets......Page 67 2.3 Continuity and differentiability......Page 69 2.4 Convex optimization......Page 73 2.5 Conjugate functionals......Page 74 2.6 Duality theory......Page 76 2.7 Applications of duality theory in a posteriori error analysis......Page 78 3. A POSTERIORI ERROR ANALYSIS FOR IDEALIZATIONS IN LINEAR PROBLEMS......Page 84 3.1 Coefficient idealization......Page 85 3.2 Right-hand side idealization......Page 108 3.3 Boundary condition idealizations......Page 117 3.4 Domain idealizations......Page 123 3.5 Error estimates for material idealization of torsion problems......Page 129 3.6 Simplifications in some heat conduction problems......Page 136 4.1 Linearization of a nonlinear boundary value problem......Page 144 4.2 Linearization of a nonlinear elasticity problem......Page 160 4.3 Linearizations in heat conduction problems......Page 177 4.4 Nonlinear problems with small parameters......Page 186 4.5 A quasilinear problem......Page 190 4.6 Laminar stationary flow of a Bingham fluid......Page 193 4.7 Linearization in an obstacle problem......Page 199 5.1 A posteriori error analysis for regularization methods......Page 210 5.2 Kacanov method for nonlinear problems......Page 220 5.3 Kacanov method for a stationary conservation law......Page 226 5.4 Kacanov method for a quasi-Newtonian flow problem......Page 236 5.5 Application in solving an elastoplasticity problem......Page 243 6. ERROR ANALYSIS FOR VARIATIONAL INEQUALITIES OF THE SECOND KIND......Page 252 6.1 Model problem and its finite element approximation......Page 254 6.2 Dual formulation and a posteriori error estimation......Page 260 6.3 Residual-based error estimates for the model problem......Page 265 6.4 Gradient recovery-based error estimates for the model problem......Page 272 6.5 Numerical example on the model problem......Page 279 6.6 Application to a frictional contact problem......Page 288 REFERENCES......Page 304 Index......Page 318 This work provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear var- tional problems. An error estimate is called a posteriori if the computed solution is used in assessing its accuracy. A posteriori error estimation is central to m- suring, controlling and minimizing errors in modeling and numerical appr- imations. In this book, the main mathematical tool for the developments of a posteriori error estimates is the duality theory of convex analysis, documented in the well-known book by Ekeland and Temam ([49]). The duality theory has been found useful in mathematical programming, mechanics, numerical analysis, etc. The book is divided into six chapters. The first chapter reviews some basic notions and results from functional analysis, boundary value problems, elliptic variational inequalities, and finite element approximations. The most relevant part of the duality theory and convex analysis is briefly reviewed in Chapter 2. "This volume provides a posteriori error analysis for mathematical idealizations in modeling boundary value problems, especially those arising in mechanical applications, and for numerical approximations of numerous nonlinear variational problems. The author avoids giving the results in the most general, abstract form so that it is easier for the reader to understand more clearly the essential ideas involved. Many examples are included to show the usefulness of the derived error estimates. This volume is suitable for researchers and graduate students in applied and computational mathematics, and in engineering."--Jacket Numerical simulation/scientific computation is now playing a more and more important role, and has become one of the three basic tools in science and technology, in addition to experimentation and theory.