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

Do you remember the equation derived in our last blog post ? -

In the finite element world, K is named element stiffness matrix, M is named element mass matrix and f is called the element load vector. As promised in the last post we will analyse these matrices today starting with the element stiffness matrix. First we calculate the gradient vector of

and obtain

Therefore we end up with

To obtain the element matrices, it is necessary to evaluate the integrals just derived. Restricting ourselves to the two-dimensional case, we will now show that this can be done analytically for simple linear triangular elements where we have shown the element shape functions in a recent blog post.

As long as G is constant inside the element, the element mass matrix can be
easily evaluated

and for the load vector we obtain

This series of blog posts summarizes a chapter of the book A Workout in Computational Finance. The next blog post will discuss how to incorporate a time discretisation to our  setup. It will be published on Monday, 22nd of July.