# Finite difference/element method : time step and spatial resolution close to a finite singularity

I'm using the finite element method (FEM), but my question is quite a global question. It's related to this question but it is not the same.

Let's assume we have this equation : $$\partial_t c - u\frac{dc}{dx}=\frac{d}{dx}\Big(D(c)\frac{dc}{dx}\Big)$$

with $$u$$ a positive constant.

and with $$D(c)>0$$ and $$\lim_{c\rightarrow 1} D(c) \rightarrow{+\infty}$$

You can assume that there are Dirichlet conditions at the boundaries, for example : $$c_{|\partial \Omega}=0$$. And a Neumann boudary conditions : $$c'(x)=0$$ at the boundaries $$\partial \Omega$$.

What I'm observing is that for some reasons sometimes $$c$$ goes beyond the singularity $$c=1$$. And I don't want him to do that...

Here is a picture of the situation :

So I discretized the problem near the node $$n$$ where $$c$$ is close to 1 :

Let's first call $$c_n^t$$ the value of $$c$$ at node $$n$$ and time $$t$$. The condition for being below $$c=1$$ is therefore : $$\Delta t (u\frac{dc}{dx}+\frac{d}{dx}\Big(D(c)\frac{dc}{dx}\Big))<1-c_n^t$$. If we discretize it, we get :

$$\Delta t(u\frac{c_{n+1}^t-c_{n-1}^t}{2\Delta x} + D(c_n^t)\frac{c_{n+1}^t+c_{n-1}^t-2c_{n}^t}{\Delta x^2 }+D'(c_n^t)(\frac{c_{n+1}^t-c_{n-1}^t}{2\Delta x})^2)<1-c_n^t$$

What I'm looking for is a general result like the one for the Péclet stability, where we need $$Pe<1$$. I'm talkinf about Péclet since here too we need $$diffusion>convection$$.