I'm looking to numerically compute the solution to

$$ k(x,u) \partial_t u - \Delta u = f \quad\quad\text{ in } \Omega \times [0,T]$$ 

where $k$ is a continuous but nonlinear (in $u$) real-valued function and $\Omega \subseteq \mathbb{R}^2$. I want to use the Finite Element Method, but I cannot seem to manage the nonlinear $k$. I wrote the code myself and it already works for $k = const$, i.e. in the "standard case". 

My problem is, that I cannot get rid of the $k(u)$ factor and therefore don't reach a formula like 
$$ M \dot{u} - A u  =  F $$
like in [page 80 here][1] (mass matrix $M$, stiffness matrix $A$).


Any help or reference is appreciated.


  [1]: http://www.ima.umn.edu/~arnold//8445.f11/notes.pdf