120892864
Zur Verzweigung bei einfachen Eigenwerten
Let W and Y be real Banach spaces and F:W → Y be a smooth mapping. Let xɛW be a solution of F(z)=0, Bifurcation theory deals with the problem of determining the structure of the zeros of F near x. We give a modification of a well-known theorem of Crandall and Rabinowitz which is useful for applications, i.e. for the computation of bifurcation points. A numerical example is included.
123856837
Numerische Behandlung von Verzweigungsproblemen bei Gewöhnlichen Randwertaufgaben
We consider the numerical solution of bifurcation problems for ordinary differential equations in the case of simple eigenvalues. First we treat the case of bifurcation from the trivial solution. We also give a new method for the numerical treatment of secondary bifurcation. It consists in the accurate determination of the bifurcation point for which a convex unrestricted minimization problem is derived, and the computation of both branches by conventional methods after transforming the problem.
120311575
Multigrid Bifurcation Iteration
We analyze a multigrid technique for the numerical computation of bifurcating solutions of large, sparse systems of nonlinear parameter-dependent equations which arise from discretizations of nonlinear elliptic eigenvalue problems by finite differences or finite elements. The algorithm consists of a nested iteration procedure which employs a multigrid method for singular problems. The underlying iteration scheme related to the Lyapunov–Schmidt method has been widely used for proving the existence of bifurcating branches and also sometimes for numerical calculations. Our multi-grid technique is shown to be a very efficient and theoretically sound tool for computing bifurcating branches of solutions of nonlinear elliptic eigenvalue problems on arbitrarily bounded domains in $\mathbb{R}^2 $. Several numerical examples confirm this.
117989535
Numerical Methods for Bifurcation Problems — A Survey and Classification
The purpose of this paper is to give an account of recent developments in numerical methods for the solution of bifurcation problems. For readers not too familiar with our subject we shall first summarize important applications of bifurcation and dicuss some of the basic ideas, problems and tools of bifurcation theory.
123517753
On the numerical approximation of secondary bifurcation problems
We discuss stable numerical methods for the approximation of the solutions of a nonlinear parameter - dependent equation near a non-trivial bifurcation point. The problem of finding the bifurcation point is reformulated as a well-posed equation of higher dimension. The nearby branches can be calculated in a stable manner after applying a certain transformation having its origin in the Lyapunov — Schmidt theory. We also treat the perturbed bifurcation problem and present numerical results.
123580420
Multi-Grid Solution of Bifurcation Problems
A continuation strategy is developed for a method of inverse iteration type to compute solutions of nonlinear eigenvalue problems. The solution curves are parametrized by a norm of the solution. Thus continuation near turning points does not represent any difficulties. A multi-grid version of the algorithm is presented and successfully applied to solve problems on general two-dimensional domains. Numerical results including a comparison with a more standard approach are also given for the continuation strategy proposed for the basic algorithm.
107642364
The software factory challenge
The Eureka Software Factory project (ESF) was set up by a group of European partners in 1987. This text presents ESF's main ideas and achievements, and relates how the concepts pioneered by the project have become part of a general movement in both the the industrial and academic domains.
814779
A microprogrammed implementation of EULER on IBM system/360 model 30
An experimental processing system for the algorithmic ~anguage EULER has been implemented in microprogramming on an IBM System/360 Model 30 using a second Read-Only Storage unit. The system consists of a mlcroprogrammed compiler and a microprogrammed String Language Interpreter, and of an I/O control program written in 360 machine language. The system is described and results are given in terms of microprogram and main storage space required and compiler and interpreter performance obtained. The role of microprogramming is stressed, which opens a new dimension in the processing of interpretive code. The structure and content of a higher level language can be matched by an appropriate interpretive language which can be executed efficiently by microprograms on existing computer hardware.