A geometrical comparison between cell method and finite element method in electrostatics

Cell Method, a Finite Formulation technique, is compared in detail with the Finite Element Method (FEM), a differential-based numerical technique. In the finite formulation technique, Poisson's equation is described starting from a topological foundation. The final set of algebraic equations resulting from the two approaches are compared in matrix form. The equivalence of the coefficient matrices is proven for a Voronoi dual mesh and linear shape functions in the FEM. The difference between the source (charge) vectors in the two approaches is described. It is shown that the use of linear shape functions in the FEM is equivalent to the use of a barycentric dual mesh for charge vectors. Also, it is shown that the coefficient matrix derived from a variational technique, FEM, can be interpreted using the simple geometrical concept of a parallel plate capacitor. As an example, a Schottky barrier diode with a non-uniform doping profile is considered. The results demonstrate the differences between the two approaches in the case of a non-zero electric charge density.