@misc{DeuflhardFreundWalter1990, author = {Deuflhard, Peter and Freund, R. and Walter, Artur}, title = {Fast Secant Methods for the Iterative Solution of Large Nonsymmetric Linear Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-330}, number = {SC-90-05}, year = {1990}, abstract = {A family of secant methods based on general rank-1 updates has been revisited in view of the construction of iterative solvers for large non- Hermitian linear systems. As it turns out, both Broydens "good" and "bad" update techniques play a special role - but should be associated with two different line search principles. For Broydens "bad" update technique, a minimum residual principle is natural - thus making it theorectically comparable with a series of well-known algorithms like GMRES. Broydens "good" update technique, however, is shown to be naturally linked with a minimum "next correction" principle - which asymptotically mimics a minimum error principle. The two minimization principles differ significantly for sufficiently large system dimension. Numerical experiments on discretized PDE's of convection diffusion type in 2-D with internal layers give a first impression of the possible power of the derived "good" Broyden variant. {\bf Key Words:} nonsymmetric linear system, secant method, rank-1 update, Broydens method, line search, GMRES. AMS(MOS) {\bf Subject Classifications:} 65F10, 65N20.}, language = {en} } @misc{Bornemann1990, author = {Bornemann, Folkmar A.}, title = {An Adaptive Multilevel Approach to Parabolic Equations I. General Theory \& 1D-Implementation.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-320}, number = {SC-90-04}, year = {1990}, abstract = {A new adaptive multilevel approach for parabolic PDE's is presented. Full adaptivity of the algorithm is realized by combining multilevel time discretization, better known as extrapolation methods, and multilevel finite element space discretization. In the theoretical part of the paper the existence of asymptotic expansions in terms of time-steps for single-step methods in Hilbert space is established. Finite element approximation then leads to perturbed expansions, whose perturbations, however, can be pushed below a necessary level by means of an adaptive grid control. The theoretical presentation is independent of space dimension. In this part I of the paper details of the algorithm and numerical examples are given for the 1D case only. The numerical results clearly show the significant perspectives opened by the new algorithmic approach.}, language = {en} } @misc{Gatermann1990, author = {Gatermann, Karin}, title = {Symbolic solution of polynomial equation systems with symmetry.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-311}, number = {SC-90-03}, year = {1990}, abstract = {Systems of polynomial equations often have symmetry. The Buchberger algorithm which may be used for the solution ignores this symmetry. It is restricted to moderate problems unless factorizing polynomials are found leading to several smaller systems. Therefore two methods are presented which use the symmetry to find factorizing polynomials, decompose the ideal and thus decrease the complexitiy of the system a lot. In a first approach projections determine factorizing polynomials as input for the solution process, if the group contains reflections with respect to a hyperplane. Two different ways are described for the symmetric group Sm and the dihedral group Dm. While for Sm subsystems are ignored if they have the same zeros modulo G as another subsystem, for the dihedral group Dm polynomials with more than two factors are generated with the help of the theory of linear representations and restrictions are used as well. These decomposition algorithms are independent of the finally used solution technique. We used the REDUCE package Groebner to solve examples from CAPRASSE, DEMARET and NOONBURG which illustrate the efficiency of our REDUCE program. A short introduction to the theory of linear representations is given. In a second approach problems of another class are transformed such that more factors are found during the computation; these transformations are based on the theory of linear representations. Examples illustrate these approaches. The range of solvable problems is enlarged significantly.}, language = {en} } @misc{Deuflhard1990, author = {Deuflhard, Peter}, title = {Global Inexact Newton Methods for Very Large Scale Nonlinear Problems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-303}, number = {SC-90-02}, year = {1990}, abstract = {Newton methods for nonlinear problems are known to require the solution of a sequence of linear problems of the same type. For very large scale problems, as understood herein, the arising linear systems can only be solved by iterative methods. Then Newtons iteration appears as outer iteration. The question of interest will be to control the accuracy of the inner iteration such that the convergence speed of Newtons method is preserved. The purpose of the paper is to combine the concept of inexact Newton methods with the concept of the affine invariant exact Newton methods - which is important for problems with ill- conditioned Jacobian matrices (such as typical 2-D or 3-D discretized partial differential equations).}, language = {en} } @misc{WulkowDeuflhard1990, author = {Wulkow, Michael and Deuflhard, Peter}, title = {Towards an Efficient Computational Treatment of Heterogeneous Polymer Reactions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-292}, number = {SC-90-01}, year = {1990}, abstract = {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.}, language = {en} } @misc{Hoppe1989, author = {Hoppe, Ronald H. W.}, title = {Numerical Solution of Multicomponent Alloy Solidification by Multi-Grid Techniques.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-285}, number = {SC-89-10}, year = {1989}, abstract = {The solidification of an \$ N \$-component alloy is described by an initial boundary value problem for a system of degenerate parabolic equations modelling heat conduction and mass diffusion. Discretizing implicitly in time and by piecewise linear finite elements in the space variables, at each time step the solution of a system of quasivariational inequalities is required. For the numerical solution of that system, a multi-grid algorithm is developed by making use of game theoretic concepts and duality arguments from convex analysis. Finally, the efficiency of the algorithm is demonstrated by displaying numerical results for a ternary alloy.}, language = {en} } @misc{Yserentant1989, author = {Yserentant, Harry}, title = {Two Preconditioners Based on the Multi-Level Splitting of Finite Element Spaces.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-274}, number = {SC-89-09}, year = {1989}, abstract = {The hierarchical basis preconditioner and the recent preconditioner of BRAMBLE, PASCIAK and XU are derived and analyzed within a joint framework. This discussion elucidates the close relationship between both methods. Special care is devoted to highly nonuniform meshes; our theory is based exclusively on local properties like the shape regularity of the finite elements.}, language = {en} } @misc{Maierhoefer1989, author = {Maierh{\"o}fer, Gerhard}, title = {Ein paralleler adaptiver Algorithmus f{\"u}r die numerische Integration.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-265}, number = {SC-89-08}, year = {1989}, abstract = {Der Bericht ist die Ausarbeitung eines Vortrages, der auf dem Transputer Anwender Treffen (TAT'89) im September 1989 gehalten wurde. Es wird die Parallelisierung und Implementierung eines adaptiven Algorithmus zur numerischen Integration (Romberg Quadratur) beschrieben. Ausgew{\"a}hlte Meßergebnisse sind enthalten. {\bf Keywords:} Numerischer Algorithmus, Romberg Quadratur, paralleler adaptiver Algorithmus, dynamische Lastverteilung und Prozessorzahl, lokaler Speicher, Nearest-Neighbour-Architektur, Transputer, TDS, OCCAM2.}, language = {de} } @misc{BuddeWulkow1989, author = {Budde, Uwe and Wulkow, Michael}, title = {Computation of Molecular Weight Distributions for Free Radical Polymerization Systems.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-253}, number = {SC-89-07}, year = {1989}, abstract = {Modeling of free radical polymerization leads to very large and usually stiff systems of ordinary differential equations which cannot be solved directly in an efficient way. This paper presents the application of a new approach called discrete Galerkin method to a realistic example - the polymerization of methyl methacrylate(MMA). The method is characterized by a Galerkin approximation on the basis of orthogonal polynomials of a discrete variable which represents the polymer degree. It allows the efficient computation of solutions of complete kinetic schemes with time- or moment-dependent reaction coefficients by reducing the complexity to a few differential equations. The approximation error can be controlled by an error estimation. In the case of MMA polymerization a reduction of computational effort by a factor of about 25 compared to a standard method can be obtained for the quasi-steady-state approximation of the model. In addition solutions of the instationary kinetic scheme can be easily computed.}, language = {en} } @misc{HuamoShuli1989, author = {Huamo, Wu and Shuli, Yang}, title = {MmB-A New Class of Accurate High Resolution Schemes for Conservation Laws in Two Dimensions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-248}, number = {SC-89-06}, year = {1989}, abstract = {In this paper we present the MmB schemes, which preserve the local maximum and minimum bounds of the initial data in the smallest union of mesh elements of previous time step containing the domain of dependence of the solution on the mesh element with center at point \$ P \$\ under consideration. In 1-D, the MmB schemes are almost identical with TVD schemes. As well-known, there is no second-order TVD scheme in 2-D, nevertheless, we present here two classes of 2-D second-order accurate MmB-schemes. It is proved that 1-D discrete MmB (or TVD) and 1-D semi-discrete TVD schemes may have second-order accuracy at (nonsonic) critical points, but cannot be of uniformly second-order accurate in the whole neighborhood of the critical points. New accurate high resolution flux limiters are suggested. Numerical results for 1-D and 2-D test problems are given. {\bf Keywords:} Difference scheme, TVD, MmB, flux limiter.}, language = {en} } @misc{KornhuberRoitzsch1989, author = {Kornhuber, Ralf and Roitzsch, Rainer}, title = {On Adaptive Grid Refinement in the Presence of Internal or Boundary Layers.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-230}, number = {SC-89-05}, year = {1989}, abstract = {We propose an anisotropic refinement strategy which is specially designed for the efficient numerical resolution of internal and boundary layers. This strategy is based on the directed refinement of single triangles together with adaptive multilevel grid orientation. It is demonstrated by several numerical examples that compared to usual methods, the new anisotropic refinement ends up in more stable and more accurate solutions at much less computational cost. {\bf Keywords:} Adaptive finite elements, directed refinement, adaptive grid orientation, convection diffusion equation, internal and boundary layers.}, language = {en} } @misc{Moeller1989, author = {M{\"o}ller, H. Michael}, title = {Multivariate Rational Interpolation: Reconstruction of Rational Functions.}, url = {http://nbn-resolving.de/urn:nbn:de:0297-zib-227}, number = {SC-89-04}, year = {1989}, abstract = {In this paper we consider the problem of reconstructing a multivariate rational function, when only its values at sufficiently many points are known. We use for the reconstruction of bivariate rational functions a bivariate rational interpolation operator investigated by Siemaszko [7] and a new one, compare both by examples in a Computer Algebra system, and present their multivariate generalizations. {\bf Keywords:} Multivariate rational interpolation, reconstruction, symbolic computation.}, language = {en} }