Formulation of a non linear PDE in FEniCS

1

Hello everyone,

I am new on FEniCS (I discovered the FEniCS project yesterday).
I am interested in this project because I have to solve a non linear PDE. I use Python, on Windows.
I am reading the Springer tutorial on FEniCS but everything is not clear to me… It seems that FEniCS is a wonderful tool to solve my problem, but I do not understand how to use it.

I would like to have an exemple of a non linear PDE solver, on which I could base myself to solve my problem.

You will find attached below a screenshot of my problem, with the two boundary conditions.

image

I apologize in advance if my question is poorly worded or if you find my request rude…

Thank you in advance for any kind of answer,

Have a nice day

2

Assuming you already know the theory of FEMs, the nonlinear Poisson demo is a good place to start.

3

Thank you very much for your answer nate!

I’ve just read the nonlinear Poisson demo. To be honest, I do not know the theory of FEMs.

I notice that in my problem, I only have one dimension, and a time-dependant parameter (whereas in the nonlinear Poisson demo there are two dimensions and no time-dependent parameter). Does it mean that the mesh I will use only needs to be a one dimensional mesh ?

4

One spatial dimension typically means you need a one dimensional mesh (a discretised interval) with a standard FEM.

It will be worth reading the standard texts on finite element methods before jumping in the deep end with your problem.

There is a tutorial with FEniCS here.

5

Thank you again nate!

I read the tutorial you gave me. So, to summary, if I find the variational formulation of my problem, I will be able to find the value of V(A,t) for every t in [0, T], given the path of “A” ? (“A” follows a geometric brownian motion, so in my mind, I woukd like to simulate the path of “A”, for t in [0, T], with T a given parameter, and then find the value of V(A,t))

I’ve got a question about the tutorial. I do not totally understand the “Choosing a test problem” part, is this part only here to confirm that the variational formulation of the nonlinear Poisson problem is good?

6

As I said before, I strongly advise you get a solid understanding of the fundamentals of finite element methods before embarking on solving your stochastic PDE with FEniCS.

7

I understand, unfortunately, since I am not given much time to do this work, I cannot go deeply in the theory. Is there maybe another solution to my problem / a way to obtain the variational formulation of my problem using another method ?

8

As I’m sure you’ve already investigated, for Black-Scholes type problems, there are several commercial software options.

9

In fact, for Black-Scholes type problems there are lots of solution, however my problem is not really a Black-Scholes PDE… That’s why I would like to find a way to solve it. I thank you for your advice!