Dietrich Braess Bücher

InhaltsverzeichnisI Einführung.II Konforme Finite Elemente.III Nichtkonforme und andere Methoden.IV Die Methode der konjugierten Gradienten.V Mehrgitterverfahren.VI Finite Elemente in der Mechanik elastischer Körper.Literatur.

The initial investigations into nonlinear approximation problems were conducted by P. L. Chebyshev, whose contributions are foundational to the theory of uniform approximation. His ideas paved the way for the development of best uniform approximation theories for rational functions and polynomials within a cohesive framework. The distinction between linear and rational approximation emerged in the 1960s, alongside the exploration of alternative approaches to nonlinear approximation. The introduction of tools such as nonlinear functional analysis and topological methods revealed that linearization alone is inadequate for fully addressing nonlinear families. Notably, the use of global analysis and the examination of flows within the family of approximating functions brought forth innovative concepts previously absent in approximation theory, which have since proved significant across various analytical fields. Additionally, techniques developed for nonlinear approximation are frequently applicable to linear approximation problems. A key example is the resolution of moment problems through rational approximation. The pursuit of optimal quadrature formulae or the quest for the best linear spaces often necessitates the consideration of spline functions with free nodes. The appendix of this work addresses the renowned challenge of best interpolation by polynomials.

InhaltsverzeichnisA mixed variable finite element method for the efficient solution of nonlinear diffusion and potential flow equations.Two multi-level algorithms for the dam problem.Multi-grid eigenvalue computation.Multigrid solution of the steady Euler equations.Analysis of a SOR-like multi-grid algorithm for eigenvalue problems.A multigrid treatment of stream function normal derivative boundary conditions.A multigrid method for solving the biharmonic equation on rectangular domains.A fast solver for the Stokes equations using multigrid with a UZAWA smoother.Calculations of transonic flows around single and multi-element airfoils on a small computer.Basic smoothing procedures for the multigrid treatment of elliptic 3D-operators.A preconditioned conjugate residual algorithm for the Stokes problem.List of participants.

The book offers a comprehensive introduction to finite element methods, emphasizing updated content for research and practical applications. It features an in-depth discussion on saddle-point problems and nonstandard applications, alongside a complete examination of locking phenomena in elasticity. The author thoroughly addresses the numerical solution of elliptic partial differential equations within the framework of Sobolev spaces. It serves as an essential resource for graduate students lacking a strong background in differential equations, particularly in connecting mathematics and engineering through solid mechanics.

Blending Approximations with Sine Functions.- Quasi-interpolation in the Absence of Polynomial Reproduction.- Estimating the Condition Number for Multivariate Interpolation Problems.- Wavelets on a Bounded Interval.- Quasi-Kernel Polynomials and Convergence Results for Quasi-Minimal Residual Iterations.- Rate of Approximation of Weighted Derivatives by Linear Combinations of SMD Operators.- Approximation by Multivariate Splines: an Application of Boolean Methods.- Lm, ?, s-Splines in ?d.- Constructive Multivariate Approximation via Sigmoidal Functions with Applications to Neural Networks.- Spline-Wavelets of Minimal Support.- Necessary Conditions for Local Best Chebyshev Approximations by Splines with Free Knots.- C1 Interpolation on Higher-Dimensional Analogs of the 4-Direction Mesh.- Tabulation of Thin Plate Splines on a Very Fine Two-Dimensional Grid.- The L2-Approximation Orders of Principal Shift-Invariant Spaces Generated by a Radial Basis Function.- A Multi-Parameter Method for Nonlinear Least-Squares Approximation.- Analog VLSI Networks.- Converse Theorems for Approximation on Discrete Sets II.- A Dual Method for Smoothing Histograms using Nonnegative C1-Splines.- Segment Approximation By Using Linear Functionals.- Construction of Monotone Extensions to Boundary Functions.