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

Elements II

In the last blog post we started discussing general properties of elements and took a look at a simple one dimensional element. Today we will analyse elements for a 2D setup for which we introduce the triangular element families. In both cases we focus on linear and quadratic shape functions, which we consider sufficient for financial modeling.

The two-dimensional linear triangular element is also known as simplex element

and is represented by a linear polynomial in x and y (the superscript (e) denotes an element):

The nodal values can again be used to determine the coefficients by solving a linear system (a little homework)


and an unknown quantity can again be expressed using the nodal values and the shape functions.

Omitting the superscript (e), the coefficients a, b, and c int the shape functions can be easily calculated using the node co-ordinates

Note that these shape functions satisfy the properties discussed for the one dimensional case: Each shape function has a value of one at its own node and a value of zero at the other two. Summing up all three functions always amounts to one. The shape functions vary linearly along the edges between its node and the other two nodes, and the shape function is zero along the edge opposite its node. Also note that derivatives with respect to the spatial coordinates are constant within an element.

In our book (see link below) we also present the rectangular element family and discuss numerical integration techniques used for higher order and isoparametric elements.

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

The sixth post of the series will be published on Thursday, July 4 and will show how the 

assembling of a linear system of equations using the discrete finite elements works out. We show this procedure in a step-by-step manner by applying the finite element method to a two dimensional static diffusion-reaction equation.