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In order to face the rapidly increasing need for computational resources of various scientific and engineering applications one has to think of new ways to make more efficient use of the worlds current computational resources. In this respect, the growing speed of wide area networks made a new kind of distributed computing possible: Metacomputing or (distributed) Grid computing. This is a rather new and uncharted field in computational science. The rapidly increasing speed of networks even outperforms the average increase of processor speed: Processor speeds double on average each 18 month whereas network bandwidths double every 9 months. Due to this development of local and wide area networks Grid computing will certainly play a key role in the future of parallel computing. This type of distributed computing, however, distinguishes from the traditional parallel computing in many ways since it has to deal with many problems not occurring in classical parallel computing. Those problems are for example heterogeneity, authentication and slow networks to mention only a few. Some of those problems, e.g. the allocation of distributed resources along with the providing of information about these resources to the application have been already attacked by the Globus software. Unfortunately, as far as we know, hardly any application or middle-ware software takes advantage of this information, since most parallelizing algorithms for finite differencing codes are implicitly designed for single supercomputer or cluster execution. We show that although it is possible to apply classical parallelizing algorithms in a Grid environment, in most cases the observed efficiency of the executed code is very poor. In this work we are closing this gap. In our thesis, we will - show that an execution of classical parallel codes in Grid environments is possible but very slow - analyze this situation of bad performance, nail down bottlenecks in communication, remove unnecessary overhead and other reasons for low performance - develop new and advanced algorithms for parallelisation that are aware of a Grid environment in order to generelize the traditional parallelization schemes - implement and test these new methods, replace and compare with the classical ones - introduce dynamic strategies that automatically adapt the running code to the nature of the underlying Grid environment. The higher the performance one can achieve for a single application by manual tuning for a Grid environment, the lower the chance that those changes are widely applicable to other programs. In our analysis as well as in our implementation we tried to keep the balance between high performance and generality. None of our changes directly affect code on the application level which makes our algorithms applicable to a whole class of real world applications. The implementation of our work is done within the Cactus framework using the Globus toolkit, since we think that these are the most reliable and advanced programming frameworks for supporting computations in Grid environments. On the other hand, however, we tried to be as general as possible, i.e. all methods and algorithms discussed in this thesis are independent of Cactus or Globus.