A Posteriori Error Analysis Via Duality Theory: With Applications in Modeling and Numerical Approximations
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 (). 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.
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00 otherwise adapted mesh assume assumption auxiliary function bilinear form boundary condition boundary value problem conjugate function consider constraint set convergence convex functions corner domain defined denote derive a posteriori dual problem duality theory elliptic variational inequality energy function error bound Figure finite element method finite element solution fvdx fvdx+ g2vds Gateaux derivative gradient recovery type Hence Hilbert space idealized Jn Jn Jn Jr2 Kacanov iterates Kacanov method Lax-Milgram Lemma Lemma linear problem linearized elasticity Lipschitz continuous Lipschitz domain mathematical minimization problem model problem nodes nonlinear problem normed space numerical results obstacle problem obtain parameter piecewise polynomial posteriori error analysis posteriori error estimates quantity recovery type estimator residual type estimator results based sequence side Sobolev spaces solving subset torsion problem triangle uniform mesh unique solution upper bound variational inequality weak formulation