Hi,
I am also new to deal.ii. I tried solving a non-linear equation in which I passed a vector, which just contained values of solution vector at the Gauss points.
You can easily do so by using
FEValues<dim> fe_values(fe, quadrature formula, <all flags here>).
Then, use the get_function_values to get the nodal solution vector local to the cell. You can send this vector to the function for calculating the values of coefficient.
I hope this helps.
-Rajat