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So, thank you very much for this nice introduction.

I'm surprised that you digged out my diploma thesis.

In fact, I missed a chance when leaving this field

because methods which have been developed there are now very hot in encryption.

So, if you have elliptic curve encryption methods,

they are very heavily based on algebraic geometry methods now.

So, maybe I shouldn't have done this and I would have been on a much more hot topic today.

So, who knows?

Okay, so when I thought about what I should do, thank you very much again for inviting me.

So, when Peter Knamler came to the Weierstrass Institute,

he became the head of the numerical analysis group.

And at that time, this position was less attractive than it is now

because it wasn't connected with the professorship.

And so, basically it was somehow clear that he just took the next step in his career.

So, he left us.

But what basically stayed at Weierstrass was that he introduced me into porous media

and porous media modeling, which I used in several aspects later on in my work.

But then I thought, okay, about what should I speak here?

Basically, you already said that there is also an electric side of Peter Knamler.

And we will talk about this a little bit more.

Okay, so the talk will be about ion transport without electroneutrality assumption.

Basically, this is relevant in a number of applications, for instance, colloidal systems or in clay.

This is where the interest seems to come from here in this chair.

Microstructures of porous electrodes and nanopores of polymer electrolyte membranes and fuel cells

basically are actual topics that I'm interested in.

And for instance, it's a very hot topic also in life sciences.

So, if you want to model ion channels and bio membranes, you use similar models.

And so, it's basically a very interesting topic with a broad potential of applications.

There are even more interesting questions where you could use these kinds of models, which I don't mention here.

Okay, so I'm a numerical guy mainly.

So, I want to use basically recent work on modeling, in particular on modeling,

which takes into account the finite size of ions, which incorporates the knowledge from semiconductor device simulations.

So, we have there basically the working horse as a numerical scheme is the Schaffer-Dagumme scheme,

which is unconditionally stable and guarantees mass conservation maximum principle,

which is very balanced in the sense that it is consistent with the spirit equations for thermodynamic equilibrium.

And this is basically something one wants to carry over also to this new, not completely new,

but I would say something about this modeling, which should be applied to the models,

the improved Nernst-Planck Poisson models.

Okay, so what we are talking about.

So, we have a mixture of N species, and N minus 1 species of them are ions, which are charged.

The charged number of an ion is the set alpha. This is just some integer number plus,

it could be positive, negative and so on.

If charged ions are somehow around, they set up an electrostatic potential, which follows the Poisson equation.

So, we have here this Q is, so I forgot to write it on the slide.

So, that's basically the sum of the ion concentrations times the charge numbers.

And this defines basically the source of an electrostatic field, which is characterized by the electrostatic potential.

And for the motion of ions, there are a number of driving forces.

So, one driving force is obviously the biorhythmic velocity of the mixture.

We will ignore this later on for our discussion.