Apart from its conceptual simplicity, the finite difference method has a number of appealing
features. The regularity of the grid is responsible for a regular structure of the system
matrix, allowing highly efficient linear solvers to be used. The disadvantage of this method
is also clearly visible. The introduction of grid lines in regions of strong variation of the
quantities introduces lots of grid points in regions where they are not needed. This is even
more of a problem in the three-dimensional case. Also, the orthogonal grid has little
geometrical flexibility. Surfaces which are not parallel to the grid lines are not resolved
properly unless a large number of grid points is employed.
In order to describe the non planar surfaces originated from, for example silicon oxidation,
mapping techniques have been proposed . An irregular physical domain is
transformed into an orthogonal computational domain, suitable for discretization with the
finite difference method. Such a transformation usually requires the actual device boundary
lines to be described by analytical or interpolating expressions.
M. Gritsch: Numerical Modeling of Silicon-on-Insulator MOSFETs PDF