Define functions used in weak form

variational formulation
1

Hi, I am trying to define a function that would be in the weak form in FEniCS.

However the function’s derivative is quite complicated, I am wondering if there is any good way of defining it FEniCS.
The first function I need to define is

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where u is the variable to be solved. (I am solving a nonlinear PDE).
I implement it in the following way
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The second function I need to define is the derivative of the above function (wrt u).
I can compute the derivative and directly define it, which would be a long expression. This is one way. However, I am wondering if there is a better way, especially in case when the derivative becomes too complicated.

E.g. In computing the right hand side f(x,y) of a manufacture PDE, we can use sympy’s functionalities to get an expression for f(x,y), convert the expression to C++ code and then define it using fenics Expression which recognizes C++ language. Is there a similar method for the problem I encountered? This situation is a bit different because I am defining a nonlinear function in the variable u.

Thank you very much!

2

You can automatically take derivatives with the diff function. The one point to keep in mind is that the variable with respect to which you differentiate must be labeled as a Variable before defining the function to be differentiated. So, in this case, it would look something like:

u = variable(u)
F_inv = Fermi_inv(u)
dF_inv_du = diff(F_inv,u)

where dF_inv_du is \frac{d\mathcal{F}^{-1}(u)}{du}.

3

Thank you! This is very helpful.