During the last years, scientific computing has become an important research
branch located between applied mathematics and applied sciences and engineering.
Nowadays, in numerical mathematics not only simple model problems
are treated, but modern and well-founded mathematical algorithms are applied
to solve complex problems of real life applications. Such applications are
demanding for computational realization and need suitable and robust tools
for a flexible and efficient implementation. Modularity and abstract concepts
allow for an easy transfer of methods to different applications.
Inspired by and parallel to the investigation of real life applications, numerical
mathematics has built and improved many modern algorithms which
are now standard tools in scientific computing. Examples are adaptive methods,
higher order discretizations, fast linear and non-linear iterative solvers,
multi-level algorithms, etc. These mathematical tools are able to reduce computing
times tremendously and for many applications a simulation can only
be realized in a reasonable time frame using such highly efficient algorithms.