Conclusion

The local refinement of finite element meshes proved to be an appropriate method to increase the accuracy and reliability of numerical simulations based on finite element methods. If the initial mesh is generated carefully $\mbox{AGM}^{\mbox{\footnotesize 3D}}$ and its new features provide reliable tools for error estimation and refinement.

An interface between $\mbox{AGM}^{\mbox{\footnotesize 3D}}$ and vecu has been created by the implementation of new commands. They save all graphical data of the finite element mesh in a format compatible with vecu and its preprocessing programs. Moreover they give access to the results obtained by vecu and prepare the mesh for further calculations.

The simulation of a hard magnetic cube served as a test for both the program and the technique itself. While the new commands showed their reliability, the program and the technique remain to be ``refined''. Currently finite element meshes are available only for cubic domains. Yet, $\mbox{AGM}^{\mbox{\footnotesize 3D}}$ is supplied with several error estimators and flexible input- and output-routines which need no modification if new domains are added.

Another problem is the fast growing number of elements and vertices if the grid is refined. As a consequence the calculations become very complex and slow. Up to around 2,000 elements results can be obtained within reasonable time. If the grid is refined another time it may end up with some 10,000 elements. However, as computing power and speed are improving, these limits are pushed forward. Alternatively special computers may be used. Finite element methods require the solution of large systems of linear equations. They can be solved very efficiently with vector computers which are optimized for the manipulation of vectors and matrices. The NEC SX4 B/2 at the Centre for Computing Services at the Technical University Vienna is a good example for this type of machines. It is ranked among the 20 fastest vector computers in the world.

Another problem which has been mentioned in [Bagn91] may occur. The refinement takes place in areas of major interest such as domain walls. It should improve accuracy and reliability of the simulation though it should not influence the sample and its properties. However, it has been observed that domain walls may move from finer areas to coarser areas of the grid. This behaviour can be explained by the exchange energy, which is in a finer grid different from that in a coarser grid and therefore does influence the calculations.

During the simulation of the hard magnetic cube it was not possible to verify these facts but further research work will have to take them into consideration.