The finite element method is used for solving partial differential equations (PDE). Solutions are acheived by either eliminating the differential equation completely (steady state problems), or rendering the PDE into an equivalent ordinary differential equation, which is then solved using standard techniques such as finite differences, etc.
Finite element methods have also been developed to solve integral equations such as the heat transport equation.
The method was introduced by Richard Courant to solve torsion on cylinder. Courant's contribution was evolutionary, drawing on a large body of earlier results for PDEs developed by Rayleigh, Ritz and Galerkin. Development of the method began in earnest in the middle to late 1950s for airframe and structural analysis, and picked up a lot of steam at Berkeley in the 1960s for use in civil engineering. The method was provided with a rigorous mathematical foundation in 1973 with the publication of Strang and Fix's The Finite Element Method.
Finite element methods are used in a wide variety of engineering disciplines. In computer graphics, radiosity algorithms are finite element methods.