Hows, Whys and Wherefores of FEM in Quantitative Finance - I

An Introduction to the Finite Element Method
This is the first part of a concise series of blog posts dealing with the Finite Element Method in quantitative finance. Today's blog is essentially a short introduction to the method and an outline of the basic steps necessary for application of the method to practical problems. Each step will be discussed in a blog entry.

The finite element Finite Elements Method 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.

The basic strategy of the finite element method is to divide the region of interest into smaller parts (finite elements), and to approximate the solution in each element by a simple function. The method can thus be seen as a piecewise approximation. Polynomial-type interpolation functions are most widely used for the element solutions. Choosing polynomials over other kind of interpolation functions is convenient because of two reasons:

(1) It is relatively easy to formulate and compute the finite element equations.
(2) It is possible to obtain higher accuracy by increasing the order of the polynomial.

The practical application of the finite element method can be broken down into the following series of basic steps:

(1) Discretiztion of the region of interest: The most important concepts used for grid generation, such as refinement strategies and adaptivity, will be presented during this blog series. This step also includes numbering the nodes and specifying their corresponding coordinate values.
(2) The trial function h(x) is specified (for instance, the order of a polynomial approximation) and for each element, the partial differential equation is written in terms of the unknown nodal values.
(3) Using the Galerkin method (will be discussed in the next part of the blog series) for each element a system of equations is developed. These equations are then assembled into the global system of equations that covers the whole domain.
(4) Add the boundary/terminal/interface conditions
(5) Solve the system of equations for different methods to solve linear systems of equations

The second post of the series will be published on Friday, June 7 and will give an overview over different weighted residual methods used in the formulation of the finite element equations.