As already mentioned the finite element method requires a discretized
domain. By the discretization the two-dimensional domain
is divided into elements.
Often the subdivision is treated as a preprocessing task to the
finite element technique provided automatically for different
and arbitrary shaped simulation
regions [43,44] from specific software.
In this work the elements are triangles. The elements must
not overlap and there must not be any gaps between them. A proper
discretization avoids elements, which have a small inner angle (narrow
elements). Such elements usually cause a larger simulation error.
It can be shown that the relative deviation of the simulation result
from the exact solution is inversely proportional to the sine of the
smallest angle in the triangular element [45]. Thus,
the best case will be,
if all generated triangles were equilateral. Another important
property of the discretization is the element size. Generally,
smaller elements lead to higher precision of the numerical results.
Otherwise, generation of smaller elements means generation of more
elements, which gives more unknowns and a larger linear equation system,
respectively. Thus longer simulation times and a higher memory demand
must be taken into account. A good approach is to use smaller elements
in the regions where it is expected that the field quantities will
vary intensely. In the remaining regions, in which little variations
are assumed, larger elements can be used. Such a method optimizes
the number of elements and unknowns, respectively, for a desired solution
accuracy. A number of algorithms and efficient techniques are developed
to automate the mesh generation. Finite element mesh adaptation processes
and mesh improvement techniques using posteriori error estimator guarantee
the quality of the resulting mesh [46,47].