Sapere aude! Immanuel Kant (1724-1804) Numerical simulations playa key
role in many areas of modern science and technology. They are necessary
in particular when experiments for the underlying problem are too
dangerous, too expensive or not even possible. The latter situation
appears for example when relevant length scales are below the
observation level. Moreover, numerical simulations are needed to control
complex processes and systems. In all these cases the relevant problems
may become highly complex. Hence the following issues are of vital
importance for a numerical simulation: - Efficiency of the numerical
solvers: Efficient and fast numerical schemes are the basis for a
simulation of 'real world' problems. This becomes even more important
for realtime problems where the runtime of the numerical simulation has
to be of the order of the time span required by the simulated process.
Without efficient solution methods the simulation of many problems is
not feasible. 'Efficient' means here that the overall cost of the
numerical scheme remains proportional to the degrees of freedom, i. e.,
the numerical approximation is determined in linear time when the
problem size grows e. g. to upgrade accuracy. Of course, as soon as the
solution of large systems of equations is involved this requirement is
very demanding.