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 guess it is not possible to use a FEM approach in this case?

Any help or reference is appreciated.

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.

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 $A$).

Any help or reference is appreciated.

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 guess it is not possible to use a FEM approach in this case?

Any help or reference is appreciated.