A guide to the Boundary Element Method and Green's functions for students and professionals: history, ideas, and applications

The world is very complicated. A goal of the physical sciences is to study and understand this complexity. A particularly useful tool for the mathematical description of complex physical phenomena has been differential equations. Differential equations describe the state of a physical system using limited information about the relations between the local rates of variable physical quantities. The quantities (and the state of the system) are found after the differential equation is solved. In some cases, the solutions can be found analytically, but in most cases, it must be done numerically using various computational tools, e.g. the Finite Difference Method or the Finite Element Method.

(BEM) is also a computational tool for solving a large class of differential equations, but the way it works is somewhat counterintuitive. Instead of operating with local quantities, it computes the solutions to so-called boundary integral equations. These equations give information about systems in a way that is "dual" to the way differential equations do. They describe what is happening at a particular point in a system given information about what is happening at its boundary.

The idea that the surface may store the information on the space "beneath" it seems to be attractive even to physicists. They speculate that "... everything in the Universe... even space itself is just a projection of the information stored on some distant two-dimensional surface...", just as the following movie suggests:

Watch The Fabric of the Cosmos: What Is Space?  

So, does the surface have complete information on "what lies beneath" it, just as we have seen in the movie? Not exactly, some key information is still missing. This information is the so-called fundamental (singular) solution of the differential equation, or Green's function named after George Green.

But what does George Green have to do with integral equations and numerical methods? He died in 1841, before the foundations of the theory of linear integral equations were laid by Volterra and Fredholm, and long before the idea of a computer ceased to become a subject of science fiction. However, the life of George Green was as singular and his contribution to science was as fundamental as the solution that bears his name