The FEM is a numerical method for solving partial differential equations (PDEs), and has become particularly popular in engineering and physics.
More recently also the quantitative finance community has gained interest to apply this method to the various PDEs arising in the modelling of financial instruments - it is one method of choice in the pricing and calibration engine of UnRisk from the beginning (at a time when it was not common at all in quantitative finance circles).
It is not applicable to every financial model and if, you need to use it with care - in this article it is outlined that mean reverting models for interest rates tend to become numerically difficult in regions sufficiently far away from the mean-reverting level (in the convection-reaction-diffusion PDE, convection dominates the diffusion). We apply streamline diffusion techniques to obtain stable numerical schemes.
In Mathematics of our UnRisk Insight blog you will find a series of posts about FEM (and other mathematical approaches and problems) in quant finance.
They are compilations of chapters of the book A Workout In Computational Finance that has been officially released by Wiley recently.
The workout is built of inspiring sessions about how to enjoy the freestyle of quant work, but staying strong, agile and balanced. It is about the power of mathematica schemes computing financial behavior, traps of wrong method traps and the fitness killer of sloppy implementations.
The owners of then book will be entitled to use a web page, inviting them to play around with methods discussed in the book.