11th GAMM-Seminar Kiel on
Numerical Treatment of Coupled Systems
January 20th to 22nd, 1995.
->[GAMM-Homepage]
Information about the corresponding proceedings
Reference
W. Hackbusch, G. Wittum (Eds.): Numerical Treatment of Coupled Systems.
Proceedings of the 11th GAMM-Seminar Kiel, January 20 to 22, 1995.
Notes on Numerical Fluid Mechanics Vol. 51. Vieweg-Verlag, Braunschweig,
Wiesbaden, Germany, 1995. ISBN 3-528-07651-8.
Foreword
The GAMM Committee for ''Efficient Numerical Methods for Partial
Differential Equations'' organizes seminars and workshops on subjects
concerning the algorithmic treatment of partial differential
equations. The topics are discretisation methods like the finite
element and the boundary element method for various type of
applications in structural and fluid mechanics. Particular attention
is devoted to the advanced solution methods.
The series of such seminars was continued in 1995, January 20-22,
with the 11th Kiel-Seminar on the special topic
Numerical Treatment of Coupled Systems
at the Christian-Albrechts-University of Kiel. The seminar was attended
by 100 scientist from 9 countries. 23 lectures were given, including
two survey lectures.
Different kinds of couplings are considered in this volume. The
coupling of different components may occur in the physical model. On
the other hand, a coupling of subsystems can be generated by the
numerical solution technique. General examples of the latter kind are
the domain decomposition (see p. 128) or subspace decomposition (p.
117). The local defect correction method couples different
discretizations of the same problem in order to improve the results,
although the basic linear system to be solved remains unchanged (p.
47). In general, the aim of the numerical coupling is to make use of
(efficient) subsystem solvers (p. 1). The combination of different
discretization techniques is mentioned on page 59.
Various examples of coupling different physical components are
discussed: the interaction of solid and fluid (p. 96, p. 175, and p.
198), the microelectronic systems (p. 11 and p. 86), and the coupled
modelling in groundwater flow (p. 106 and p. 139).
A topic where the coupling may be of physical and numerical nature is
the finite element boundary element coupling (FEM-BEM). This very
interesting problem is discussed in four contributions (p. 25, p. 37,
p. 73, p. 151).
Kiel, May 1995,
W. Hackbusch, G. Wittum
Contents
- ST. ARTLICH, W. MACKENS:
Newton-coupling of fixed point iterations, pp. 1
- H.-J. BUNGARTZ, ST. SCHULTE:
Coupled problems in microsystem
technology, pp. 11
- C. CARSTENSEN, D. ZARRABI:
An a posteriori error estimate for the
unsymmetric coupling of FEM and BEM, p. 25
- M. FEISTAUER, G.C. HSIAO, R.E. KLEINMAN:
On numerical treatment
of coupled BEM and FEM for nonlinear exterior problems, pp. 37
- P.J.J. FERKET:
Coupling of a global coarse discretization and
local fine discretizations, pp. 47
- B. HEINRICH:
The Fourier-finite-element method for elliptic problems
in axisymmetric domains, pp. 59
- B. HEISE:
Parallel solvers for coupled FEM-BEM equations with
applications to non-linear
magnetic field problems, pp. 73
- R. HIPTMAIR, R.H.W. HOPPE, B. WOHLMUTH:
Coupling problems in
microelectronic device simulation, pp. 86
- F. IHLENBURG, CH. MAKRIDAKIS:
Error estimates of Galerkin FEM for a
system of coupled Helmholtz equations
in one dimension, pp. 96
- H. KASPER, G. KOSAKOWSKI, T. TANIGUCHI, W. ZIELKE:
Coupled geometric
modelling for the analysis of groundwater flow
and transport
in fractured rock, pp. 106
- E. KATZER:
Subspace decomposition methods for solving the
Euler equations, pp. 117
- A. KLAR:
Domain decomposition schemes and coupling conditions
for kinetic
and hydrodynamic equations, pp. 128
- O. KOLDITZ, R. RATKE, W. ZIELKE, H.-J. DIERSCH:
Coupled physical
modelling for the analysis of groundwater systems, pp. 139
- H. SCHIPPERS, J.A. WENSING:
Coupling of boundary and finite elements
in aeroacoustic calculations, pp. 151
- W. SCHMID, F. WAGNER:
Numerical solution of the neutron diffusion
equation - adaptive concepts
in time and space, pp. 163
- U. SPECHT, CH. DICKOPP, J. BALLMANN:
A numerical scheme for
stress waves at a fluid-solid interface, pp. 175
- C. VUIK, A. SEGAL:
Solution of the coupled Navier-Stokes equations, pp. 186
- D. WEISS, G. LUBE:
Numerical simulation of temperature distribution
and seam forming
in narrow gap welding, pp. 198
Collection of Abstracts
ST. ARTLICH, W. MACKENS:
Newton-coupling of fixed point iterations
To solve a coupled system of two PDEs it may be intended not to use the
Newton-Raphson-method, for example due to the non-sparsity of the Jacobian of
the entire system or because there exist solvers for subsystems.
For this type of problems we present an iterative Newton type method which
requires only a finite number of iterations for the single equations in each
step. The algorithm is based on a formal Block Gauss elimination of the full
Newton system and the solution of the resulting Schur complement equation by
a Bi-CGSTAB-iteration. No computation of the Jacobian of the whole system is
necessary. In contrast to alternative approaches, the two sets of equations
resulting from the discretization of the two PDEs may be of the same size.
The above method had to be developed to solve a system of PDEs describing the
behaviour of a chemical reactor for the combustion of coal. This model consists
of an enthalpy balance of convection-diffusion-type and two mass balances, one
of them of reaction-diffusion-type, the other one an ODE. For the latter there
exists an analytical solution depending on the unknowns of the two other
balances.
The numerical results presented deal with this combustion model.
H.-J. BUNGARTZ, ST. SCHULTE:
Coupled problems in microsystem
technology
In microsystem technology, the numerical simulation of coupled problems
is one of the principal challenges. There are three main reasons for the
fact that, here, the coupling of different physical effects (structural
dynamics, fluid dynamics, heat transfer, or electromagnetics, e.g.) is more
frequently encountered than in the macro world: First, aspects of scaling
often lead to a dominance of surface effects on volume dependent effects.
Second, especially in sensors a lot of different physical phenomena are
used for signal conversion, and, finally, in some microsystems different
physical effects have an influence on each other.
We present a classification of the most important couplings in the
microsystem world, and we give a survey on existing solution
techniques with emphasis on methods based on the so-called partitioned
solution. Here, there is no joint model, neither continuous nor discrete,
but the coupled problem is solved by an outer iteration realizing the
coupling and by arbitrary inner solution processes for each single problem.
The coupling is done via changed boundary conditions,
geometries, or parameters after each step of iteration. This approach
seems to be advantageous, since its modularity allows the use of
existing and efficient codes for each sub-problem. Therefore, only the
outer iteration has to be organized with some kind of interface for the
coupling. Furthermore, this technique seems to be perfectly suited for
parallelization, especially for the use of (heterogeneous) workstation
clusters.
Finally, some first numerical results concerning the simulation of a
micro-miniaturized two-valve membrane pump are presented.
C. CARSTENSEN, D. ZARRABI:
An a posteriori error estimate for the
unsymmetric coupling of FEM and BEM
Some engineering applications are concerned with
a nonlinear material behaviour in a bounded domain $\Omega$
and a linear elastic material behaviour in the surrounding
unbounded exterior domain $\Omega_c$, e.g.,
in a tunnel problem, in crack problems or in Kirsch's problem.
The paper is concerned with such interface problems where
the exterior problem is rewritten via boundary integral operators.
We give an overview on certain nonlinear materials
in $\Omega$ where convergence of Costabel's symmetric coupling
and a priori error estimates can be proved. Then we focus
on a priori error estimates for the direct coupling
(where no a priori estimates are available) and for the symmetric method.
Numerical examples are given.
M. FEISTAUER, G.C. HSIAO, R.E. KLEINMAN:
On numerical treatment
of coupled BEM and FEM for nonlinear exterior problems
The contribution presents main results (obtained in cooperation with G.
C. Hsiao and R. E. Kleinman) of the investigation of the
coupled BEM and FEM applied to a nonlinear exterior boundary value
problem. The problem consists of a nonlinear partial differential
equation considered in an annual bounded domain and the Laplace
equation outside. These equations are equipped with boundary and
transmission conditions. The problem is reformulated in a weak sense
and combined with an integral equation. The attention is paid to the
solvability of this problem and of the corresponding discrete finite element -
boundary element problem, the convergence of approximate solutions to
an exact one and to a sufficiently robust solver of the discrete
coupled nonlinear problem. Computational results will also be presented.
P.J.J. FERKET:
Coupling of a global coarse discretization and
local fine discretizations
In elliptic boundary value problems with strong local variations in the solution, a basic global discretization can be improved significantly by introducing a local discretization defined in a subdomain. The local defect correction iteration [1] is based on the combination of a local and a global discretization method. We present theoretical insights assessing that the approximation obtained after two iteration steps is much more accurate than the approximation obtained after only one iteration step and that often the use of more than two iterations is not efficient. Also we will explain that in many cases a suitable choice for the "overlap" parameter $d$ in the local defect correction method is $d=0$. Numerical examples confirm the theoretical results.
The limit value of the local defect correction iteration with $d=0$ can be written as exact solution of a discrete problem on the composite grid that is composed of the global coarse grid and the local fine grid. This composite grid problem can be solved by the fast adaptive composite grid method [2]. We will show that, under reasonable assumptions, the local defect correction iteration with $d=0$ yields the same results as the fast adaptive composite grid iteration.
- Hackbusch, W.: Local defect correction method and domain decomposition techniques
, Computing, Suppl 5 (1984), pp 89-113.
- Mc Cormick, S.F.: Multilevel adaptive methods for partial differential equations , SIAM, Philadelphia, 1989.
B. HEINRICH:
The Fourier-finite-element method for elliptic problems
in axisymmetric domains
The paper deals with the Fourier--finite--element method for solving elliptic
problems in three--dimensional axisymmetric domains and gives some survey of
this approach to the Poisson equation and to the Lam\'e equations. The method
combines the approximating Fourier method with the finite--element method and
reduces the approximate solution of a problem in 3D to the solution of coupled
or even decoupled systems of elliptic problems in 2D. Algorithmic aspects of
this method, its parallelization and basic results of the numerical analysis of
this method, particularly for the Poisson equation in domains with edges, are
discussed. The results are illustrated by numerical examples.
B. HEISE:
Parallel solvers for coupled FEM-BEM equations with
applications to non-linear magnetic field problems
An efficient parallel algorithm for solving nonlinear boundary value problems
arising, e.g., in magnetic field computation, is presented. It is based on both
the idea of ''nested iteration'' (also called Full Multilevel Method) and the
parallel domain decomposition (DD) linear solvers suited for computations on
MIMD computers with local memory and message-passing principle. It makes use of
the parallel data structure of these solvers, the linearization is done by
Newton's method, the linear system is solved by CG with DD preconditioning.
The DD approach allows to couple Finite Element and Galerkin Boundary Element
Methods in a unified variational problem. In this way, e.g., magnetic field
problems in an infinite domain with Sommerfeld's radiation condition can be
modelled correctly. The problem of a nonsymmetric system matrix due to
Galerkin-BEM is overcome by transforming into a symmetric but indefinite matrix
and applying Bramble/Pasciak's CG for indefinite systems. For preconditioning,
the main ideas of recent DD research are applied.
Test computations on a Parsytec transputer based system were performed for both
model problems and a real-life electric motor. The very good scale-up when
using $16 \cdots 128$ processors for problems of growing size (100.000 $\cdots$
800.000 unknowns) shows the high efficiency of the proposed algorithm.
First tests were run on the Parsytec Xplorer and GC-Power Plus systems. They
have shown a reduction of calculating time by a factor of about 1/29.
R. HIPTMAIR, R.H.W. HOPPE, B. WOHLMUTH:
Coupling problems in
microelectronic device simulation
The cost-effective design of electronic microstructures requires an advanced
modeling and coupled simulation of various physical effects. The classical
isothermal approach leads to the basic drift-diffusion model for semiconductor
device simulation. In the stationary case, it represents a coupled nonlinear
system consisting of a Poisson equation for the electric potential and two
continuity equations for the electron and hole flow. We discuss the pros and
cons of various discretization schemes that have to take into account extreme
layers in the solution caused by the nature of the coupling. We further address
efficient numerical solution techniques including adaptive multilevel methods.
Finally, to allow for nonisothermal and magnetic effects we consider extensions
of the classical model and we discuss perspectives for their appropriate
numerical treatment.
F. IHLENBURG, CH. MAKRIDAKIS:
Error estimates of Galerkin FEM for a
system of coupled Helmholtz equations in one dimension
The quality of discrete solutions to the reduced wave equation depends on the
relation between the physical parameter (frequency) and the numerical parameters
(stepsize, order of polynomial approximation). For a one-dimensional model
problem of fluid-solid interaction we analyze continous and discrete stability
(inf- sup-condition) and prove an error estimate for a Galerkin FEM
(h-p-method). The estimate holds in the asymptotic range while practical
computations are carried out for preasymptotic values of h. Reference to
recently proven statements for the uncoupled problem and discussion of
computational results for the model problem are employed to address the error
behaviour in the preasymptotic range.
H. KASPER, G. KOSAKOWSKI, T. TANIGUCHI, W. ZIELKE:
Coupled geometric
modelling for the analysis of groundwater flow and transport
in fractured rock
A geometric modelling method is proposed for the finite element analysis of
groundwater flow in 3-dimensional rock, which is separated by an arbitrary
number an location of fracture planes. The method prepares coupled meshes using
hexahedral elements for the rock matrice, quadrilateral elements for the
fracture planes and line elements for the boreholes. The mesh generation method
is based on the Delaunay triangulation, and the efficiency of the proposed method
is examined through several test problems, for examples Grimsel and Asp\'o
sites. Using these numerical models the groundwater flow is analyzed. As the
result it has been proved, that the modelling method is valid for flow problems.
However, hexahedral elements with an improved shape are required for solving
transportation problems in fractured rock. Therefore several theoretical
proposals to generate meshes with improved geometric properties are included in
this paper. The numerical solution shown in this investigation are obtained
using the solver {\sc Rockflow}, which is developed at the Institut f\"ur
Str\"omungsmechanik und elektrisches Rechnen im Bauwesen, Universit\"at
Hannover.
E. KATZER:
Subspace decomposition methods for solving the
Euler equations
A robust subspace decomposition method for solving first order systems of
partial differential equations is presented. The algorithm is based on a
carefully constructed sequence of four subspaces. Parallel additive and
sequential multiplicative subspace correction methods are applied.
The discretisation is based on a vertex centered finite volume method for a
triangular grid. The discrete system is obtained by minimizing the residuals of
the discrete flux equations.
We obtain robust methods with uniformly bounded asympotic error reduction rates.
Convergence rates are independent of grid size and only slightly dependent of
parameters of the elliptic and hyperbolic model system.
Convergence rates for the elliptic case are of order 0.1 for isotropic and 0.4
for anisotopic cases. For the isotropic elliptic case, error reduction rates
are similar to results of standard twogrid methods. In the hyperbolic cases,
convergence is slightly larger and of order 0.8 but still uniformly bounded for
all characteristic directions. A combination of the subspace decomposition
approach as preconditioner for a conjugate gradient method improves the
convergence speed.
The approach opens the way to fast solutions of the Euler equations which
comprise a nonlinear system of composite elliptic/hyperbolic type.
A. KLAR:
Domain decomposition schemes and coupling conditions
for kinetic and hydrodynamic equations
For numerical calculations in rarefied gas dynamics domain decomposition methods
involving kinetic and aerodynamic equations have become of considerable
interest. Two main problems in the coupling procedure are the statement of the
correct coupling condition at the interface between the kinetic and aerodynamic
domain and the development and investigation of suitable algorithms to obtain
the solution of the DD - problem.
The correct choice of the coupling conditions depends on the physical situation
at the interface between the two types of equations. Assuming thermal
equilibrium there, the so called Marshak conditions, based on the equality of
fluxes, are appropriate. If we consider instead of equilibrium situations
nonequilibrium states at the interface, then the above coupling conditions will
not lead to the correct results. Here the matching can be done by modelling the
interface region by a transition layer. Scaling the space coordinate in the
layer one obtains a kinetic linear half space problem, leading to the correct
coupling conditions. The validity of the obtained coupled solution can be
proven in this case. To make the approach reasonable from a numerical point of
view we developed a fast numerical scheme for the half space problem, based on a
Chapman Enskog type expansion procedure. In a special case this method is shown
to be equivalent to the well known so called variational methods.
The algorithm to obtain finally the solution of the DD - problem is usually an
iterative procedure solving in turn the kinetic and aerodynamic equations in
their respective subdomains, a variant of the well known Schwarz alternating
method. We consider here the linearized versions of the above equations and
prove convergence of the alternating scheme.
O. KOLDITZ, R. RATKE, W. ZIELKE, H.-J. DIERSCH:
Coupled physical
modelling for the analysis of groundwater systems
Salt-water intrusion and hydrothermal convection are classic examples of
density-dependent problems in groundwater hydrology. The economic importance of
these phenomena has invited continuous research in the fields of aquifer
contamination/remediation, waste disposal, and geothermal energy.
Especially if considering coupled nonlinear problems the numerical models have
to be carefully verified. The commonly used benchmarks for testing
density-dependent simulations are the Henry Problem and the Elder Problem. We
calculate these benchmarks using different codes (ROCKFLOW-DM2, FEFLOW). The
topic of our first study is to show the impact of different numerical schemes
(explicit and implicit in time) and equation solvers (direct and iterative) on
the accuracy of results and on the computation time consumed.
Based on the verified numerical models more complex problems are tackled. In
this way mixed convection in geothermal fractures including thermoelastic
effects due to the permanent heat minig is simulated. Furthermore, first
results of modelling thermohaline processes are presented. For this purpose the
flow must be coupled with both, mass and heat transfer simultaneously.
H. SCHIPPERS, J.A. WENSING:
Coupling of boundary and finite elements
in aeroacoustic calculations
A coupled computational system of finite elements and boundary elements is
presented to estimate scattering and refractive effects on the acoustic pressure
waves through a boundary layer surrounding an aircraft's fuselage. The noise
source due to isolated propellers or propfans is assumed to be known. The
fuselage is modelled by an infinitely long cylinder with a constant cross
section in axial direction. The propagation effects through the boundary layer
can be significant and should be included in the noise predictions. Due to the
boundary layer, the incident acoustic field will be refracted.
The acoustic pressure in the boundary layer is modelled by linearizing the Euler
equations about a unidirectional cylindrically sheared mean flow. The governing
equation is solved in the frequency domain using piecewise linear finite
elements on a triangular grid. The total pressure field is assumed to consist
of a scattered field and an incident field given by a plane wave, propagating
harmonically with frequency $\omega$ in time and with wavenumber $\alpha$ in
axial direction. Outside the boundary layer the pressure satisfies the
convective Helmholtz equation with a uniform wavenumber. Here the scattered
pressure is given by a boundary integral representation formula which involves
as unknown distributions the pressure and its normal derivative at the edge of
the boundary layer. This formula follows from applying Green's theorems to the
Helmholtz equation. The pressure at the edge is approximated by piecewise
linear boundary elements, whereas the normal derivative is modelled by piecewise
constant boundary elements. The boundary element grid coincides with the outer
boundary of the finite element grid.
The diffraction problem is investigated for a cylinder with a cross section
similar to the Fokker 50 aircraft fuselage.
W. SCHMID, F. WAGNER:
Numerical solution of the neutron diffusion
equation - adaptive concepts in time and space
An important aspect in the safety analysis of nuclear power plants is the
computation of the distribution of the neutrons in the reactor. A widely used
mathematical model is given by the energy-dependent diffusion equation.
Integrating over a finite number of energy intervals results in the so-called
multigroup diffusion equations representing a coupled system of parabolic
differential equations where the coupling is due to scattering effects.
We discretize implicitly in time and by mixed finite elements on rectangular
geometries in three space dimensions and present an adaptive solution concept
featuring combined time step control and grid refinement techniques. We discuss
in detail the proper coupling of the energy levels as well as the basic tools
for realizing the adaptive solver.
U. SPECHT, CH. DICKOPP, J. BALLMANN:
A numerical scheme for
stress waves at a fluid-solid interface
The lecture deals with a numerical scheme for stress waves at the interface of
an inviscid, compressible fluid and an elastic-plastic solid. The problem is to
couple domains of solution of non-linear, hyperbolic PDE's with
different types of waves and wave speeds.
The scheme couples a FE-solver for the Euler equations and a Godunov method for
elastic-plastic wave propagation in solids proposed by Lin and Ballmann [1]. It
is based on the scheme at the boundary for the solid and a bicharacteristic
formulation for the fluid. Instead of integrating along the characteristic
lines belonging to the speed of sound of the fluid, the integration is carried
out along the characteristic directions belonging to the fastest wave speed of
the solid, according to the von Schmidt wave effect. An iterative procedure
applying the contact conditions to the interface leads to a convergent scheme.
As a test example demonstrating the efficiency of the numerical scheme the
pressure produced by a collapsing bubble is prescribed as initial condition for
the fluid.
- X. Lin and J. Ballmann. Numerical method for elastic-plastic waves in cracked
solids, part 2: Plane strain problem. Archive of Applied Mechanics, 63:283--
295, 1993.
C. VUIK, A. SEGAL:
Solution of the coupled Navier-Stokes equations
Discretization of the incompressible Navier-Stokes equations leads to large
linear systems of equations of the following form \begin{equation} \left(
\begin{array}{cc} A & B\\ C & 0\\ \end{array} \right) \left( \begin{array}{c}
u\\ p\\ \end{array} \right) = \left( \begin{array}{c} f_u\\ f_p\\ \end{array}
\right). \label{een} \end{equation} One of the difficulties to solve this
system is the zero block at the lower right corner. Many different techniques
are used to circumvent these difficulties. In most approaches the velocity and
pressure equation are decoupled, e.g. pressure correction, artificial
compressibility, penalty methods etc. In this paper we solve the original
system (\ref{een}) using iterative methods of Krylov subspace type. In general
the Krylov subspace methods for non symmetric system, GMRES, GCR, Bi-CGSTAB etc.
can be applied to (\ref{een}). However in general the rate of convergence of
the un-preconditioned Krylov subspace methods is very slow. This together with
the fact that the spectrum of the coefficient matrix used in (\ref{een})
contains eigenvalues with positive and negative real parts motivates the search
for good preconditioners. We use a preconditioner based on an incomplete LU
decomposition. Due to the zeroes on the main diagonal it is possible that the
ILU decomposition breaks down (zero pivot elements). To prevent such a break
down special ordering techniques are used. We give results using this
preconditioner combined with various Krylov subspace methods. At present, for
2D f.e.m. discretized problems direct methods are hard to beat, however for 3D
f.e.m. discretized problems the preconditioned iterative methods are superior
with respect to memory requirements and CPU-time.
D. WEISS, G. LUBE:
Numerical simulation of temperature distribution
and seam forming in narrow gap welding
The topic of this paper is the numerical simulation of temperature distribution
and weld pool deformation during narrow gap arc welding. Knowledge of the
temperature field and the seam geometry is the basis of process optimization (in
particular to ascertain the influence on the fusion and solidification as well
as on the change in structure.)
A strongly coupled nonlinear problem has to be considerd. The energy transport
is described by a convection dominated diffusion-convection equation
(Fourier-Kirchhoff model). The deformation of the pool surface is determined by
minimization of the surface energy which is influenced by the arc pressure,
gravitation and the surface stress. An improved model of the arc intensity
distribution is introduced instead of the standard Gaussian distribution in
order to study heavily deformed molten pools during the joining of plates in the
range of medium sized and thick metal sheets in different positions.
The coupled non-linear system is linearized by the simple iteration method. The
singularly perturbed character of the problem is taken into account by using a
stabilized Galerkin finite element method of Galerkin/Least-squares type. In
particular, we study the robustness of the numerical method with respect to
different aspects of the model. Numerical results for several test problems
will be given.
How to contact the authors by email
Artlich, Stefan artlich@tu-harburg.d400.de
Bungartz, Hans-Joachim bungartz@informatik.tu-muenchen.de
Carstensen, Carsten carstensen@mathematik.th-darmstadt.de
Ferket, P.J.J. peterf@win.tue.nl
Feistauer, M. feist@karlin.mff.cuni.cz
Heise, Bodo heise@miraculix.numa.uni-linz.ac.at
Heinrich, Bernd heinrich@mathematik.tu-chemnitz.de
Hoppe, R.W.H. rohop@mathematik.tu-muenchen.de
Ihlenburg, Frank ihl@olgao.umd.edu
Katzer, Edgar edgar.katzer@mathematik.uni-magdeburg.d400.de
Kasper, Harald kasper@appel012.hydromech.uni-hannover.de
Klar, Axel klar@mathematik.uni-kl.de
Kolditz, O. kolditz@appel012.hydromech.uni-hannover.d400.de
Lube, Gert lube@namu13.gwdg.de
Makridakis, Ch. makr@athina.edu.uch.gr
Schippers, H. schipiw@nlr.nl
Schulte, Stefan stefan.schulte@zfe.siemens.de
Specht, Ulf usp@leibniz.lufmech.rwth-aachen.de
Vuik, C. c.vuik@math.tudelft.nl
Wagner, Frank wagnerf@mathematik.tu-muenchen.de
Zarrabi, Darius zarr@ifam.uni-hannover.d400.de
Jens Burmeister,
Lehrstuhl Praktische Mathematik, Mathematisches Seminar II,
Christian-Albrechts-Universität Kiel, Hermann-Rodewald-Str. 3, 24098 Kiel
Tel.: 0431-880-4462, Fax: 0431-880-4054,
Email: jb@numerik.uni-kiel.de
[Last modified: Mon May 13 11:58:44 MET DST 1996]