The essentials of the finite element method to solve differential equations: an illustrative case in physics

Abstract

Here we review the basic steps of the modern and accurate finite element method in its application to solve ordinary differential equations in physics. To exemplify this well-in-vogue numerical technique, we have chosen the second order Poisson-Boltzmann equation, which is a classic equation of colloid science. Aiming to formulate a viable, but didactic, implementation of the finite element technique, we have combined a linear basis of functions, the Galerkin weighted residuals method, the Swartz-Wendroff approximation and the Picard iteration algorithm. In summary, the finite element method transforms a differential equation into a simpler system of algebraic equations for the coefficients of the approximate solution in terms of a set of basis functions. We describe the full computational realisation of the finite element procedure and, also, we examine the corresponding Poisson-Boltzmann numerical predictions for various representative conditions.