**History**

George Orwell, the famous English writer, is attributed with the following quote: "At age 50, every man has the face he deserves." The same could be said about scientific theories and methods. The boundary element method is now about 50 years old and is in an excellent shape; its popularity is growing and its areas of applications have never been broader.

It all started in the 1960s. By then, interest in integral equations "became to change, even to fade.". The reasons for this are described in the paper of Maurice Jaswon, a pioneer in advocating the use of boundary integral equations and boundary elements for their numerical treatment. Jaswon wrote

"Integral equations afford a powerful means of attacking the boundary-value problems of potential theory and classical elasticity, but since Hilbert's days have not been greatly exploited. Several reasons for this can be suggested. Most integral equations of physical significance involve singular, or weakly singular, kernels, thereby hampering the procedures of both theoretical and numerical analysis. Also, the effective numerical solution of an integral equation inevitably means having to solve a fairly large number of simultaneous linear algebraic equations, a problem within the capacity of only recently available digital computers. The third difficulty is of an analytical nature, namely, that of ascertaining whether or not any given integral equation possesses a solution."

It was clear from the abstract that practical applications were the main targets.

"This paper makes a short study of Fredholm integral equations related to potential theory and elasticity, with a view to preparing the ground for their exploitation in the numerical solution of difficult boundary-value problems."

This and two more papers (http://rspa.royalsocietypublishing.org/content/273/1353/237.full.pdf+html, http://rspa.royalsocietypublishing.org/content/275/1360/33.short) by Jaswon and his co-workers prepared the ground but it was not broken yet. It was young engineers, not mathematicians, who really did it. As with George Green long ago, they also were not concerned that some rigorous mathematical foundations (error estimates, etc.) of the new numerical method were still to be developed. This is how Thomas Cruse, one of the pioneers of the method, describes the moment of truth related to the results later published in the seminal paper by Frank Rizzo:

"I had the distinct personal privilege to have been in the outer office when Frank came bounding in to declare, “It works!"

The rest is history….To find more about the early history of the BEM see the following references and the bibliography therein.