FEM for a nonlinear parabolic PDE

Question

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 (mass matrix $M$, stiffness matrix $A$). I also can't seem to reach any formulation which would allow me to apply the Newton method or something similar. Is it even possible to use a FEM approach in this case?

Any help or reference is appreciated.

Answer 1

As a first shot, I suggest you use Rothe's method (time discretization first), rather than the method of lines (space discretization first).

If you employ implicit Euler for that, at every time step $l$, you will have to solve the system

$$ -\tau \Delta u^l+k(u^l)u^l = k(u^{l-1})u^{l-1} + \tau [ \Delta u^{l-1} + f^l], $$

which is a semilinear elliptic problem. Then you can apply, e.g., Newton or fixed point iteration.

Answer 2

You will have to iterate out the problem. After discretization, you get a problem of the form $$ M(U^n) U^n + \Delta t \; A U^n = F^n(U^{n-1}) $$ where the mass matrix depends on the solution $U^n$ of the n-th time step. This nonlinear system of equations has to be solved by an iteration, e.g., the Newton method.

An example for a different question, but using the same idea, is given here: https://www.dealii.org/developer/doxygen/deal.II/step_15.html