This is a very contentious issue, one that academics love to debate over a cool long-neck of a friday evening. I am going to stick my head on the block here & try to explain the difference, happy chopping my academic friends.
The terms ‘finite element method’ & ‘finite element analysis’ seem to be used interchanably in most documentation, so the question arises is there a difference between FEM & FEA ??
The answer is yes, there is a difference, albeit a subtle one that is not really important enough to loose sleep over.
The finite element method is a mathematical method for solving ordinary & elliptic partial differential equations via a piecewise polynomial interpolation scheme. Put simply, FEM evaluates a differential equation curve by using a number of polynomial curves to follow the shape of the underlying & more complex differential equation curve. Each polynomial in the solution can be represented by a number of points and so FEM evaluates the solution at the points only. A linear polynomial requires 2 points, while a quadratic requires 3. The points are known as node points or nodes. There are essentially three mathematical ways that FEM can evaluate the values at the nodes, there is the non-variational method (Ritz), the residual mehod (Galerkin) & the variational method (Rayleigh-Ritz).
FEA is an implementation of FEM to solve a certain type of problem. For example if we were intending to solve a 2D stress problem. For the FEM mathematical solution, we would probably use the minimum potential energy principle, which is a variational solution. As part of this, we need to generate a suitable element for our analysis. We may choose a plane stress, plane strain or an axisymmetric type formulation, with linear or higher order polynomials. Using a piecewise polynomial solution to solve the underlying differential equation is FEM, while applying the specifics of element formulation is FEA, e.g. a plane strain triangular quadratic element