This audio is presented by the University of Erlangen-Nürnberg.

This is a presentation of the two necessary and typical steps of analysis.

Three, stability, which is often underestimated in its importance.

Consistency, this we do explicit or implicit depending on the type of the method.

And finally, of course, convergence and order of convergence.

And what we have seen now on the one hand side, if we have stability in a, say, norm

which is related to the discretization matrix in an energy norm, then we can transfer this.

We know how to describe stability of a one-step method.

And what are the consequences then for the corresponding estimate without time discretization.

So, we are now in the other version is to look at other norms, namely L infinity norms.

And this we can do by going in the direction of the validity of a maximum principle

and the various variants of it, which in the end, as we have seen already in the very beginning

with finite difference methods allows us to have stability in L infinity.

So, we are in the process of looking at that.

And I will hopefully finish this now soon and then on top of that discuss the outcome stability,

the outcome in order of convergence estimates for the different versions we have now.

I'm not sure whether we can finish this today, but we will more or less finish this

so that we still have two dates then left.

So, either we have to finish this aspect, but at least we have some time left

to look at least at very simple versions of nonlinear problems.

So, of course we have to be aware that we are very much in the realm of toy problems at the moment.

And usually problems one wants to solve in reality look a little bit more complex

in very, very, very many aspects.

But of course we have to set a foundation.

Maybe in this conjunction I should point out that there are two courses next semester

which might be or hopefully are of interest for you.

That is the continuation of this course.

So, there is a part two which is again if it is demanded from the audience it will be in English.

The major context is to go now a little bit away from the so to speak

to more diffusive dominated processes, more and more to convective dominated processes.

And we know already that the finite element analysis and also the finite element idea

comes then a little bit to its end.

We are then coming in the situation that the alpha in the ellipticity estimate goes to zero.

Of course that would mean we have all our estimates which we have

but they would all have to be astronomical constants in front of that.

And in numerical practice this means that these methods in the way we have it right now are of no use anymore.

They are still convergent methods but of course nobody can afford to really compute up to the domain of convergence.

So we need other and there is also of course a certain clear physical background behind of that.

Our discretizations are undirected in some sense.

They are based on central differences but convective processes are directed.

And if you remember we have seen already one,

the most simple and most important queue in conjunction of finite difference methods

where we said okay we have this convective part.

It is maybe a good idea, it seems to be a good idea to take the central difference quotient

because it has second order but it turns out that then because of this undirected nature

for example we need further strong assumptions to preserve the good matrix properties.

The matrix properties underlying the maximum principle.

This led to conditions with the Peclet number, with the Cell Peclet number.

The Cell Peclet number has to be small but if the Peclet number of a problem

which measures the relationship between convection and diffusion is big, say 10,000