Introduction to Numerical Electrostatics Using MATLAB

7Summary and Overview

7.1 Where We Were, Where We’re Going

This very short chapter marks a break, ending the discussion of the method of moments and beginning discussion of the finite difference (FD) and finite element (FE) methods. This is a good place to compare and contrast these methods.

All three methods are approaches to solving for the electrostatic variables charge and voltage in a given geometric structure of electrodes and possibly dielectric interfaces. They are all based on developing a set of linear algebraic equations that approximate continuous variables with approximate locally defined variables.

The method of moments (MoM) takes as its solution domain all of (three-dimensional) space. Charge is constrained to exist on conducting electrodes. These electrodes can have thickness (i.e., they themselves can be three-dimensional bodies), but since the laws of electrostatics guarantee that all the charge will move to the surface of three-dimensional (3d) electrodes, it doesn’t matter whether if the electrodes have thickness or are simply arbitrarily shaped thin skins. This is very important mathematically, because even though we are dealing with 3d space, our variable — the charge — exists only on two-dimensional (2d) surfaces.

We divided the electrode surfaces into either square or triangular planar regions, or cells, each of which is assumed to have a uniform charge density. Since the cells each have a finite area (not necessarily all the same) and a linear equation ...