Compare to the finite difference method typically used for solving differential equations, which analyses local behavior. The finite element method models the area to be solved as a patchwork of linear or low-order areas; then, the long-term behavior within each area can be solved as a system of equations over the behavior of all areas. Used in many fields within engineering to model complex behaviors.
Example from Wikipedia: left has the elements (creating a mesh); the behavior within each mesh is solved with a system of equations.