Black-Scholes coupled system

In the previous post we have seen a simple example of the Black-Scholes equation, where a single option have been considered. What happens in the case where more than options have to be studied at the same time? Well, simple extend the model to many variablesas following:

The coefficients σ, μ and r are of course different between the two equations. However, one can simply raise the question what happens when a coupling of the two dependent variables takes place. At the beginning we can assume a simple linear coupling of the two prices which will then lead to the Black-Scholes system:

As we can see now both prices u1 and u2 appear in both equations. This means that we need some robust and at the same time flexible numerical method for the solution of the above system, which leads us of course to the finite element method.