Finite Element Method

In the late 18th century, mathematicians such as Lagrange recognized
that a variety of physical phenomenon were governed by partial
differential equations (PDE). 'Finite Element Method' (FEM) is one of the
central techniques using partial differential equations and is applied to a
variety of problems in physics simulation or to study engineering
components.

The generally acknowledged originator of FEM was the German
Mathematician, Richard Courant who used this technique in 1943 to
solve a torsion problem on a cylinder. Actually, he referred to this
technique long back in 1922 in his book on function theory. Courant, of
course, drew heavily from the work of stalwarts like Lord Raleigh, Boris
Galerkin, Walter Ritz etc.

The main driving force behind developing FEM was the insolubility of
PDEs for all but simple geometries. FEM is basically a transformation to
discretize the system. The target region is divided into a number of
geometrical elements. In each element the continuous field of PDE
variables is modeled by a local polynomial approximation controlled by
a few coefficients.

Then all these elements are linked by the values at shared nodes. This
results in a set of simultaneous algebraic equations which can be solved
numerically by optimization techniques and matrix algorithms.