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

Elements I
In today's blog post we start giving an overview over different element and the shape functions Ni(x) used for generating the trial functions h(x). Independent of the dimension we first list a number of general properties valid for all shape functions we will discuss:

(1) A shape function is always one at its designated node (same nodal index) and zero on the other nodes of the element

(2) at any point in the element (including the boundaries) the sum over all shape functions is equal to one

Many problems in computational finance can be described by one-dimensional models, e.g., instruments under one factor short rate models or all options with only one underlying. The one-dimensional region is then a line and the division into elements or subregions is very straightforward. The concept of interpolation of a certain order between nodal values we show in the following for the one dimensional case can easily be applied across orders and dimensions.

1D linear Element:

A line element with two end nodes Xi and Xj (Xi < Xj) has the length L=Xj - Xi and corresponding nodal values Φi and Φj. Assuming a linear interpolation such that the unknown function Φ(x) varies linearly between the nodes within the actual element we obtain: 

 The coefficients α1 and α1 can be determined from the nodal values by solving:

Such equations, where the nodal values are multiplied by affine linear functions of x, are typical for the finite element formalism. The linear functions appearing in the equations above are called shape functions or interpolation functions. In the literature, the variable N with a subscript that indicates the node of the element is frequently used to denote shape functions. 

Often vector notation is used to display the equations.

The figure below displays the shape functions Ni and Nj:

It is easy to see, that the general properties listed at the beginning of the post are fulfilled by our shape functions.

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

The fifth post of the series will be published on Thursday, June 27 and will show how to apply the concept we derived in today's post can be extended to higher orders and higher dimensions.