Infinite Element Methods - Ying, Lung-An

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Résumé :
As its name indicates, in the infinite element method the underlying domain is divided into infinitely many pieces. This leads to a system of infinitely many equations for infinitely many unknowns; but these can be reduced by analytical techniques to a finite system when some sort of scaling is present in the original problem. The simplest illustrative example, described carefully at the beginning of the first chapter of the book, is the solution of the Dirichlet problem in the exterior of some polygon. The exterior is subdivided into annular regions by a sequence of geometrically expanding images of the given polygon; these annuli are then further subdivided. The resulting variational equations take the form of a block tridiagonal Toeplitz matrix, with an inhomogeneous term in the zero component. Various efficient methods are described for solving such systems of equations. ... The infinfte element method is, whereever applicable, an elegant and efficient approach to solving problems in physics and engineering. Professor Yings welcome book makes it available to the community of numerical analysts and computational scientists. (From the Preface by Peter D. Lax)Die Infinite Elemente Methode wird haupts?chlich bei der Berechnung singul?rer L?sungen partieller Differentialgleichungen und bei deren L?sung von Gleichungen auf unbeschr?nkten Gebieten angewandt. In dem Buch wird ein spezielles numerisches Verfahren vorgestellt, das bisher noch relativ unbekannt geblieben ist.

Sommaire:
Contents: Chapter I: Two dimensional exterior problems of the Laplace equation - Fourier methods - Iterative methods - General elements - Three dimensional exterior problems of the Laplace equation - Problems on other unbounded domains - Corner problems - Nonhomogeneous equations and nonhomogeneous boundary conditions - Plane elasticity problems - Calculation of stress intensity factors - Exterior Stokes problems - Non-similar problems - Chapter II: Foundations of Algorithm - Infinite element spaces - Shift matrices - Further discussion for the infinite element spaces and the shift matrices - Shift matrices for the plane elasticity problems - Combined stiffness matrices - Structure of general solutions - Block circular stiffness matrices - Iterative method of the first type - Iterative method of the second type - General elliptic systems - Exterior Stokes problems - Nonhomogeneous equations and the Helmholtz equation - Chapter III: Some auxiliary inequalities - Approximate properties of piecewise polynomials - H1 and L2 convergence - Maximum priniciple and uniform convergence - A superconvergence estimate - Term by term convergence near the corner - Chapter IV: Boundary value problems and Eigenvalue problems - Stress intensity factors - Stokes external flow - Navier-Stokes external flow.