High-Precision Finite Difference Method Calculations of Electrostatic Potential

The finite difference method (FDM) is one of the standard methods [1–7] for electrostatic potential calculations in nonanalytic geometries. In this technique, typically, a single, square mesh is overlaid upon the geometry and subsequently relaxed. During the relaxation, the potentials at successive points within the mesh are evaluated using an appropriate algorithm, itself being a function of the potentials at the surrounding mesh points. The process is continued until there is no further change in the potential at any of the points within the mesh upon subsequent iterations through the mesh.

The above process while being relatively simply is remarkably inaccurate. To achieve reasonably high precisions, a very closely spaced mesh must be used that can take a very long time to relax. Early efforts to improve the precision for a given mesh were directed to the area of algorithm development and were to a large extent disappointing [8, 9]. The resultant precisions seemed unfortunately to be reasonably independent (within a factor or two) of the algorithmic precision (see Fig. 30.3 of [9].)