The finite element method is one of the most powerful numerical methods available for solving partial differential equations; which apply over complex shapes. Very often, in engineering science, it is difficult to solve a partial differential equation, which applies over a complicated shape. The process therefore is to sub-divide the complex shape into lots of simpler shapes, on which the complex differential equation can be solved. The process then is to solve the complex differential over each simpler shape and 'join' all the simpler shapes together, ensuring compatibility and equilibrium at the inter-element boundaries. This often results in thousands of simultaneous equations, which can be solved on a digital computer. Writing of the computer program is 'relatively simple', compared with solving such a difficult mathematical problem by traditional methods. The method can be used for structural analysis, dynamics, vibrations, fluid flow, thermodynamics, acoustics, electrostatics, magnetostatics, seepage through porous media, electrical and fluid networks, electronics, etc., etc. Large commercial computer packages are available, such as PAFEC, ANSYS, NASTRAN, ABACUS, LUSAS, etc., etc; which make solution relatively simple for most problems in engineering.
For more information, consult:

1) Ross, C.T.F, (1996) "Finite Element Techniques in Structural Mechanics", Woodhead Publishers, Cambridge, UK.
2) Ross, C.T.F, (1998) "Advanced Applied Finite Element Methods", Woodhead Publishers, Cambridge, UK.
3) Ross, C.T.F., (1996) "Finite Element Programs in Structural Engineering & Continuum Mechanics", Woodhead Publishers, Cambridge, UK.