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 : enter image description here

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 $diffusion>convection$.

Thanks in advance



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