Summary
Finite element analysis is a method for solving partial differential equations (PDEs). There are many other methods for solving PDEs with different features, but finite element methods are attractive both because they work well with complex geometries and because they are relatively simple to implement and analyze. I use existing codes where appropriate, but have released several finite element software packages for simulating problems from MEMS design and analysis of human bone.
Related
FEAP
Extensions to an academic Finite Element Analysis Program
BoneFEA
Finite element analysis of human bones