The basic idea of the finite element method is to divide the solution domain into a finite number of subdomains that are termed elements. These elements are connected only at nodal points in the domain and on the element boundaries. In this way, the solution domain is discretized and represented as a patchwork of elements. Frequently, the finite element boundaries are straight lines or planes, so if the solution domain has curved boundaries, these are approximated by a series of straight, flat segments.

The mathematical interpretation of the finite element method requires us to generalize our definition of an element and to think of elements in less physical terms. Instead of viewing an element as a physical part of the system, we view it as part of the solution domain where the phenomena of interest are occurring. We imagine the solution domain to be sectioned by lines (or general planes in n dimensions) that define the boundaries of an element. The elements are interconnected only at imaginary nodal points on the boundaries or surfaces of the elements. For solid-mechanics problems, we no longer need to imagine that the elements deform or change shape; rather, we define them as regions of space where a displacement field exists. The nodes of an element are then simply located in space where the displacement and, possibly, its derivatives are known or sought. Similarly, for fluid mechanics problems, the elements are regions over which a pressure field exists and through which the fluid is flowing.