More FEM - Time

In our series of blog posts

Hows, Whys and Wherefores of FEM in Quantitative Finance

we discussed the different tasks necessary to get a discretised version of a diffusion-reaction equation. To apply the FE method to problems arising in quantitative finance still some work remains to be done. Todays post will focus on the time discretisation of a time dependent diffusion-reaction equation which can be written in semi-discretised form as

The variation of the time derivative on the left-hand side within an element can
be stated as

In Galerkin formulation, the residual integral for this term is

where C is called the element’s capacitance matrix. We can use the finite difference method for discretising the transient terms. Using a general time stepping scheme  (for different values of Θ the time stepping scheme is implicit, explicit or semi-implicit) we obtain

Rearranging this equation yields an equation of the form UΦ= b for the nodal
values of the unknown quantity Φ:

The integral defining the capacitance matrix,

can be evaluated analytically for the linear triangular element yielding

The post excerpts a chapter of the book A Workout in Computational Finance. The next blog post will discuss how to setup the global matrices for the system of linear equations and how to incorporate boundary conditions.