The fundamental idea of the finite element method is the replacement of continuous functions by piecewise appproximations, usually polynomials. Although the finite element method itself is relatively new, its development and success expanding with the arrival and rapid growth of the digital computer, the idea of piecewise approximation is far from new. Indeed, the early geometers used ‘finite elements’ to determine an approximate value ofπ. They did this by bounding a quadrant of a circle with inscribed and circumscribed polygons, the straight-line segments being the finite element approximations to an arc of the circle.
In this way, they were able to obtain extremely accurate estimates. Upper and lower bounds were obtained, and by taking an increasing number of elements, monotonic convergence to the exact solution would be expected. These phenomena are also possible in modern applications of the finite element method. One remark regarding ancient finite elements: Archimedes used these ideas to
determine areas of plane figures and volumes of solids, although of course he did not have a precise concept of a limiting procedure. Indeed, it was only this fact which prevented him from discovering the integral calculus some two thousand years before Newton and Leibniz. The interesting point here is that whilst many problems of applied mathematics are posed in terms of differential equations,
the finite element solution of such equations uses ideas which are in fact much older than those used to set up the equations initially.