### How to compute the gradient of a bilinear form in C++?

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1
9 months ago by
Hi all,

I need to compute the gradient of a bilinear form.

A little bit of theory, in order to make you understand better.
Given a bilinear form, continuous and coercive on a Hilbert space H:

$a:HxH\rightarrow R$a:HxHR

The  Gateaux derivative of this (in a "point" u) is a linear functional defined on H such that:
$L\psi=a\left(u,\psi\right)\forall\psi\in H$Lψ=a(u,ψ)ψH

By the Riesz's theorem, there exists a representative of H, say J, called "the gradient of a(.,.)" such that:

$\left(J,\psi\right)_H=a\left(u,\psi\right)\forall\psi\in H$(J,ψ)H=a(u,ψ)ψH

Where (.,.)H is the inner product (L2 in our case) in H.

Now, when everything is discretized using FEniCS, this representative should be a dofin::Function Jh such that:

$\left[J_h\right]_i=a\left(u,\psi_i\right)$[Jh]i=a(u,ψi)

where Jhi is the ith coefficient of Jh relative to the basis function phii of the discretized space, and "a" our dolfin::Form.

Now, my question is: how do I compute this quantity? I tried to use the "NonlinearVariationalProblem.h" header, but it does not seem to have the tools we need. Also I tried some tool to extract all the basis function and compute that quantity "by hand", but I can't accomplish this task.

Community: FEniCS Project

3
9 months ago by
This should work

u = Function(V)
v = TestFunction(V)
# Define a = a(u,v)
J = assemble(a)​

Thank you.
I am coding in C++. Should this be included in the .ufl file?
written 9 months ago by Francesco Clerici
1
9 months ago by
I don't know if this is the only way, but see the following for my solution (answer at bottom) to computing the gradient of an arbitrary scalar function in Fenics (in C++).

https://fenicsproject.org/qa/10947/computing-derivatives-in-c

--JW