Beyond DIV, CURL and GRAD: modelling electromagnetic problems using algebraic topology

Abstract

Have you ever questioned why do we almost always use vector calculus and differential equations in electromagnetics? Are these the only tools at our disposal for studying and modelling electromagnetic problems? Well, the answer is no and this paper is about an alternative approach to model electromagnetic problems using algebraic topology. This approach has a few advantages compared to the familiar differential formulation-based methods in modelling electromagnetic and multiphysics problems. In differential formulation we need one of the several numerical methods like finite difference, finite element, method of moments and spectral methods to obtain approximate solution to the electromagnetic problem. In contrast, algebraic topology-based method leads directly to discrete formulation using global quantities. Furthermore, in the case of electromagnetics, all underlying global quantities are scalars. Hence, there is no need for vector calculus. In addition, we also avoid interpolating local vector field quantities as required in several numerical methods. Though algebraic topology offers powerful and elegant tools for solving various engineering problems, many applied physicists and engineers are not familiar with this subject. This is partly due to the fact that most publications on this topic burden us with complicated mathematical analysis filled with convoluted jargons. If one can introduce the key ideas of algebraic topology using familiar concepts, computational and applied physics, and engineering community can appreciate and benefit from its advantages over methods based on differential formulation. With this goal, we present the main ideas of algebraic topology applied to electromagnetics problems.