Karl Schellbach (1804–1892)—One of the fathers of the finite element method?

Abstract

AbstractCalculations by aerospace and automotive engineers in the 1950s for industrial applications are usually viewed as the beginnings of the Finite Element Method (FEM). However, some of its basic ideas can already be found on earlier occasions. One of the examples already discussed in the literature is the treatment of the brachistochrone problem by Gottfried Wilhelm Leibniz (1646–1716) in 1697, even if he did not rigorously prove his ideas there. The mathematics educator Karl Schellbach (1804–1892) appears to have provided the first proof for the brachistochrone problem using FEM‐like methods in an article of 1851. However, one has to look at the details: Schellbach certainly first solves some elementary problems of the calculus of variations in this article by discretizing the situation. But then he generalizes this method in order to derive the general Euler‐Lagrange equation. And only on the basis of this differential equation and by use of standard methods from differential calculus he deals with the problem of the brachistochrone. In the final sections of his article, Schellbach generalizes his method to higher dimensions, which makes his considerations appear even more FEM‐like. For example, in his treatment of Plateau's problem he begins by dividing the domain of definition into triangles, which is even shown in an illustration.