Comparison between an analytical method and two numerical methods on a given electrostatic potential determination problem

In solving field problems, there are mainly three types of techniques: experimental, analytical, and numerical. Experiments are expensive, time consuming, and usually do not allow much flexibility in parameter variation. In this article we will consider the last two solving methods. Analytical methods are the most rigorous ones, providing exact solutions, but they become hard to use for complex problems. Numerical methods have become popular with the development of the computing capabilities, and although they give approximate solutions, have sufficient accuracy for engineering purposes. The problem we have studied is determining the distribution of electrostatic potential inside a conducting rectangular box having one of the armatures at V₀=1V potential and the other three at 0V potential. Three methods were considered in solving the problem: the separation of variables analytical method, the deterministic finite difference numerical method, and the probabilistic Monte Carlo numerical method. The numerical algorithms were implemented in matlab program. We compared the results to the analytical method, and raised the number of iterations until we reached a 10-10V order difference between last iterations values. The grid step size was also optimized, to minimize the error of the numerical finite difference solutions. This way, the same accuracy was obtained with a much smaller number of iterations. All the results obtained with the numerical methods were compared to the exact analytical solution.