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Jahresbericht 1990
(1991)
The success of the German Reunification requires a unique migration from the traditional Eastern Block system to a modern free society. The successful migration must include a shift from a low state of the art in technology to the high level of science and technology present in the Western Community. The role of supercomputing in this transition will include leading the process of organizational change in research and development. The paper contains a short overview on the current state of this evolution, on the currently used political and administrative tools to keep this evolution on track, and on the influence on the Konrad-Zuse-Zentrum, which is, as a Berlin institution, in the geographical and cultural center of this transformation.
The breakdown voltage highly depends on the electric field in the depletion area whose computation is the most time consuming part of the simulation. We present a self adaptive Finite Element Method which reduces dramatically the required computation time compared to usual Finite Difference Methods. A numerical example illustrates the efficiency and reliability of the algorithm.
An Adaptive Finite Element Method for Convection-Diffusion Problems by Interpolation Techniques.
(1991)
For adaptive solution of convection- difussion problems with the streamline-diffusion finite element method, an error estimator based on interpolation techniques is developed. It can be shown that for correctness of this error estimator a restriction of the maximum angle is to be sufficient. Compared to usual methods, the adaptive process leads to more accurate solutions at much less computational cost. Numerical tests are enclosed. {\bf Keywords: } Adaptive finite elements, convection- diffusion equation, internal and boundary layers, streamline-diffusion. {\bf Subject Classifications:} AMS(MOS): 65N15, 65N30
A high resolution Godunov-type method is developed and applied to a two-dimensional version of the Burgers equation. On one-dimensional testproblems the scheme is second order accurate for smooth flow and resolves discontinuities sharply. On two-dimensional problems, however, large numerical oscillations are introduced when discontinuities traveling obliquely to the grid directions are present. These oscillations are caused by a Strang- type operator splitting, and cannot be eliminated using flux limiters or small timesteps. {\bf Keywords:} High resolution methods, Godunovs method, operator splitting, complex Burgers equation. {\bf AMS:} 35: A40, L45, L60, L65, L, 67. 65: M05, M10.
Mit zunehmender Dezentralisierung im Bereich des Wissenschaftlichen Rechnens steigt auch der Bedarf an Softwarequellen qualitativ hochwertiger mathematischer Algorithmen. Für den Software- und Informationsaustausch wird die Einrichtung öffentlicher, elektronischer Softwarebibliotheken vorgeschlagen. Am Beispiel der eLib des ZIB werden die notwendigen technischen Konzepte, die u.a. auch in den heutigen Gegebenheiten der technischen Kommunikation begründet sind, diskutiert. Die eLib ist fenster- und menüorientiert aufgebaut und folglich im Dialog sehr einfach zu benutzen. Sie kann aber auch mit ihrer Kommandosprache allein auf dem Wege über elektronische Post abgefragt werden. Über die eLib sind, nach dem Verbund mit den Netzbibliotheken NetLib und Reduce-NetLib jetzt etwa 6000 mathematische Programme und allgemeine Dokumente verfügbar. Das Dienstleistungsangebot der eLib setzt Beiträge aus einer aktiven Benutzergemeinschaft voraus. Es besteht die Hoffnung, daß durch den öffentlichen Softwareaustausch sich Entwickler und Anwender wechselseitig in ihrer Arbeit befruchten.
The C-implementation of KASKADE, an adaptive solver for linear elliptic differential equations in 2D, is object of a set of numerical experiments to analyze the use of resources (time and memory) with respect to numerical accuracy. We study the dependency of the reliability, robustness, and efficiency of the program from the parameters controlling the algorithm.
Part III of the paper is devoted to the construction of an adaptive FEM solver in two spatial dimensions, which is able to handle the singularly perturbed elliptic problems arising from discretization in time. The problems of error estimation and multilevel iterative solution of the linear systems - both uniformly well behaved with respect to the time step - can be solved simultaneously within the framework of preconditioning. A multilevel nodal basis preconditioner able to handle highly nonuniform meshes is derived. As a numerical example an application of the method to the bioheat-transfer equation is included. {\bf AMS CLASSIFICATION:} 65F10, 65F35, 65M50, 65M60, 65N30.
A variety of secant methods has been revisited in view of the construction of iterative solvers for large nonsymmetric linear systems $ Ax = b $ stemming from the discretization of convection diffusion equations. In the first section, we tried to approximate $ A ^{-1} $ directly. Since the sparsity structure of A- is not known, additional storage vectors are needed during the iteration. In the next section, an incomplete factorization $ LU $ of $ A $ is the starting point and we tried to improve this easy invertible approximation of $ A $. The update is constructed in such a way that the sparsity structure of $ L $ and $ U $ is maintained. Two different sparsity preserving updates are investigated from theoretical and practical point of view. Numerical experiments on discretized PDEs of convection diffusion type in 2- D with internal layers and on "arbitrary" matrices with symmetric sparsity structure are given. {\bf Key words:} nonsymmetric linear system, sparse secant method, Broyden's method, incomplete factorization.
The discrete Galerkin method developed by the authors has turned out to be an efficient tool for the computational treatment of very large scale ODE systems arising in polyreaction kinetics. Up to now, this approach has been worked out in detail for homogeneous polymer reactions. The present paper deals with one line of possible extensions of the method to the case of so-called heterogeneous processes, which may appear e. g. in smog reactions. The associated mathematical models involve reaction coefficients depending on the chain length of the reacting polymer. The herein suggested extension is worked out in some detail on the basis of the earlier paper. In addition, a numerical example describing polymer degradation is included.