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Newton-Krylov-Schwarz Methods in CFD........ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 . . . . . . . . . . . . . . . . . . . . H. DANIELS, A. PETERS: PASTIS-3D - A Parallel Finite Element Projection Code for the Time-Dependent Incompressible Navier-Stokes Equations. . . . . . . . . . 31 . . .

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The most frequently used method for the numerical integration of parabolic differential equa­ tions is the method of lines, where one first uses a discretization of space derivatives by finite differences or finite elements and then uses some time-stepping method for the the solution of resulting system of ordinary differential equations. Such methods are, at least conceptually, easy to perform. However, they can be expensive if steep gradients occur in the solution, stability must be controlled, and the global error control can be troublesome. This paper considers a simultaneaus discretization of space and time variables for a one-dimensional parabolic equation on a relatively long time interval, called 'time-slab'. The discretization is repeated or adjusted for following 'time-slabs' using continuous finite element approximations. In such a method we utilize the efficiency of finite elements by choosing a finite element mesh in the time-space domain where the finite element mesh has been adjusted to steep gradients of the solution both with respect to the space and the time variables. In this way we solve all the difficulties with the classical approach since stability, discretization error estimates and global error control are automatically satisfied. Such a method has been discussed previously in [3] and [4]. The related boundary value techniques or global time integration for systems of ordinary differential equations have been discussed in several papers, see [12] and the references quoted therein.

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Many important algorithms in fields such as image processing or physical simulation necessitate solving systems of partial differential equations. Multigrid methods belong to the fastest and most versatile numerical schemes available for this purpose. With the advent of programmable graphics processing units (GPUs) there have been numerous efforts to use their capabilities - massively parallel floating point computation and high memory bandwidth - in high performance computing. In this work, multigrid methods are evaluated for their suitability towards a GPU implementation. After some introductory material on general purpose GPU programming and a short mathematical overview, a multigrid solver for the 2D Poisson equation built on C++, OpenGL and GLSL is presented in detail. The performance of various implementation techniques is benchmarked and interpreted, a number of optimization strategies are tested and the final results are compared across different hardware platforms and to a traditional CPU-based implementation. This book is aimed at scientists and programmers interested in using the potential of GPUs to speed up their multigrid-based numerical solvers.

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Free and covered ablation. The ablation of ice that is either free and open to the air or covered with other materials (with sand, rubble or boulders; on glaciers with moraines) comes to pass by physical processes, that can be described by equations. The amount of the ablation can be calcu lated, but depends on so many parameters, that it is not possible to show all connections of the ablation processes in one single diagram. Therefore in chapter 2 andin the appendix many dia grams are shown, that represent the dependence of the ablation on the different meteorological factors and-in case of covered ablation-on the heat transmission coefficient of the covering mate rial. The differences between free and covered ablation become especially clear. From these dia grams, too, one can take the amounts of selective ablation that is caused by other processes than by the difference in the covering. By using the equation of energy balance it is possible to inter prete these processes. The third chapter shows many illustrations about the glaciers and the small glacial forms of ablation in the environs of the Mt. Everest. The phenomenons shown on these photographs be come intelligible by the physical considerations ofthe first and the second chapter. EINLEITUNG Die Abtragung von Schnee und Eis nennt man Ablation; sie erfolgt vor allem durch Schmelzen und Verdunsten.

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In this diary, Bolzano does a detailed study of the concepts of multiplication, continuity, the curvature of space curves, the congruence of spatial figures and the solution to nth degree equations. Moreover there are a number of exercises for the determination and geometric representation of plane curves, exercises on the theory of probability and an essay on physics containing commentaries and critiques of the viable explanations for physical phenomena and suggestions for alternative explanations.

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In Bolzano›s correspondence with the Royal Bohemian Society of Sciences there are, among other things, a long expert‹s report on the problem of solving algebraic equations higher than those of the fourth degree und several reports on Christian Doppler’s work in physics. In the last two essays, Bolzano puts Euclidean geometry on a topological foundation. His remarks culminate in a particular case, Jordan’s famous curve theorem, for which there are various ideas for proof in Bolzano’s manuscripts.

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The GAMM Committee for "Efficient Numerical Methods for Partial Differential Equations" organises seminars on subjects concerning the algorithmic treatment of par tial differential equations. The topics are discretisation methods like the finite element and the boundary element method for various types of applications in structural and fluid mechanics. Particular attention is devoted to the advanced solution methods. The series of such seminars was continued in 1994, January 14-16, with the 10th Kiel-Seminar on the special topic Fast Solvers for Flow Problems at the Christian-Albrechts-University of Kiel. The seminar was attended by 100 scientist from 9 countries. 23 lectures were given, including two invited survey lectures. The central topic of most of the contributions are the Navier-Stokes equations. The authors discuss robust methods, parallel implementations, defect correction techniques, adaptive methods including error estimators, domain decompositions and FEM-BEM coupling. The invited lectures concern the hierarchical multigrid method for unstruc tured grids and structured adaptive finite-volume multigrid for compressible flows.

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Free and covered ablation. The ablation of ice that is either free and open to the air or covered with other materials (with sand, rubble or boulders; on glaciers with moraines) comes to pass by physical processes, that can be described by equations. The amount of the ablation can be caleu lated, but depends on so many parameters, that it is not possible to show all connections of the ablation processes in one single diagram. Therefore in chapter 2 and in the appendix many dia grams are shown, that represent the dependence of the ablation on the different meteorological factors and-in case uf eovered ablation-on the heat transmission cueffieient of the covering mate rial. The differences between free and covered ablation become especially clear. From these dia grams, too, one can take the amounts of selective ablation that is caused by other processes than by the difference in the eovering. By using the equation uf energy balance it is possible to inter prete these processes. The third chapter shows many illustrations about the glaciers and the small glacial forms of ablation in the environs of the Mt. Everest. The phenomenons shown on these photographs be come intelligible by the physical considcrations ofthc first and the second chapter. EINLEITUNG Die Abtragung von Schnee und Eis nennt man Ablation; sie erfolgt vor allem durch Schmelzen und Verdunsten.

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A guide the advanced features of the Perl programming language covers such topics as dispatch tables, iterators, partitions, parsing, and linear equations.

Edition bilingue français-anglais

~~~~ INTRODUCTION 1. - The problem for analysis and requirements 2. - Results 3. - Comparison of computer times 4. - Conclusion. INTRODUCTION In 1981 it appeared to the organizers that a workshop to compare codes to solve the Navier -Stokes equations would be of great interest to the community of numerical analysts and users working on CFD (Computer fluid Dynamics}. A test problem would have to be selected with the following features : 1. it should be suited for all known methods (finite differences, finite elements, spectral ... ) thus the geometry should be simple. 2. it should be simple so that scientists with small computer facilities can test it and the gradients in the flow should be reasonable in size so that algo­ rithms don't blow up with a limited number of points and iterations. In practice one would have to select a stationary flow at moderate Reynolds number. 3. it should be a real flow to allow experimental ~asurements. The simplest problem used previously was the cavity flow problem ; however the singularity of the data at the boundary (point A and B on figure 1} makes it difficult for the Finite Element and Spectral methods ; furthermore it is not such an easy experiment to set up. A B .... u = 2 Flow around a cylinder, although most interesting of all, is too difficult because it is unbounded (many grid points) and very quickly unsteady {Re ~30).

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