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

Weighted residual methods
After the short introduction into the Finite Element Method in the previous post, today we want to examine the different weighted residual methods.
If an approximate solution (trial function) is substituted into a differential displaymath a residual remains, since the approximate solution does not satisfy the equation. Consider, as an example, the one dimensional second order partial differential equation

with appropriate initial/terminal/boundary conditions. Substituting y(x) by the approximative solution h(x),

yields a non-zero residual R(x) since h(x) does not satisfy the equation exactly. The weighted residual method demands that the integral over the domain over the residual times a weighting function is equal zero.

The number of weighting functions Wi(x) is determined by the number of unknown coefficients in the trial function. Different choices for the weighting functions --- which are sometimes also called testing functions --- are available. We list some of the more popular choices here:

(1) Collocation method:Dirac-delta functions are used as weighting functions,
Wi(x)=δ(x-Xi). The residual will thus vanish at the points Xi if the weighted residual equation is fulfilled.

(2) Subdomain method: the idea is to force the weighted residual to be zero not just at fixed points in the domain, but over various subsections of the domain. To accomplish this, the weight functions are set to unity (W(x)=1), and the integral over the entire domain is broken into a number of subdomains sufficient to evaluate all unknown parameters.

(3) Galerkin method: if we consider a trial function of the form

the Galerkin method uses the same functions Ni(x) for Wi(x) that are used for the approximate solutions. Under the assumption that we are looking for a function y in a function space U also the testing functions W are element of the function space U (for example polynomials of order one). This method is the most popular one and is used for solving problems with first derivative terms. We will focus on the Galerkin method exclusively throughout this blog post series.

(4) Least squares method: the residual is used as a weighting function and a new error term is defined which is then minimised with respect to the unknown coefficients of the trial functions.

This series of blog posts summarizes a chapter of the book A Workout in Computational Finance

The third post of the series will be published on Thursday, June 13 and will give an overview over different types of grids and grid generating algorithms.