So, welcome everybody to our third session of the finite element method.

First of all, I would like to come back to the issue of a possible shifting of the lecture and the tutorial.

Thank you very much for contributing here.

The result is as follows.

What is very important is that I ask for the availabilities at possible alternative dates

and also ask whether there are people who are not available at these dates.

The result is as follows.

Out of 38 responses, 6 are not available at the possible alternative dates.

Another important issue is that I also ask if the lecture or the tutorial would be shifted.

Should only the lecture or the lecture and tutorial be shifted or do you not want to shift the lecture and tutorial?

Here the result is as follows.

5 responses are that I do not want to change the initial schedule.

Since I have to assume that those people have chosen their schedule according to the announcement on Campo,

I decided not to shift the lecture and tutorial.

There will be videos available.

We start with these video takes today and they will be made available via the video portal of FAU.

In summary, we keep the dates as they are right now.

In addition, you will have the option to watch the videos of lecture and tutorial.

Yes, please.

There will be videos for both tutorial and lecture.

This was the first announcement here.

Now let's come back to our actual finite element consideration here.

First of all, is there anything that we should discuss?

Do you have any doubts concerning last time?

If this is not the case, then I would like to show you this recapitulation.

I have uploaded that on Stutton.

These recapitulations summarize individual chapters or individual parts of chapters.

The first recapitulation here is on how to arrive from a differential equation to a system of algebraic equations.

We consider a model problem where we had the second derivative of a function u plus a function g,

both functions of x, should be zero in a specific interval with given boundary conditions.

This is a linear differential equation of second order.

Our aim was to transfer that into a set of algebraic equations which can then be solved by a computer.

The first step was to set up the so-called weak form.

To this end, we identified here this sum as the residuum, and the residuum has to vanish.

We multiplied the residuum with a test function, and then we did some term manipulations.

Eventually, we arrived here at this formulation where v is the test function,

and we have a term where both the test function and the function itself appear as their first derivatives.

We have another term here. This has to be equal to zero.

This is an integral formulation here, and of course we still have the boundary conditions here.

This is very important for all test functions.

We discussed this last time why this is important.

The function spaces are given here.

Function space S for the solution and V for the test function.

Both functions have to be taken from the Sobolev space H1, and they differ in the choice of boundary conditions.

Then we introduced an approximate solution, meaning that we substituted both the test function and the function U

by an approximation which is here given as a linear combination of coefficients and so-called shape functions.

Or you can also call them ansatz functions.

Based on this, we rewrote the weak form and we identified that now we have to require that this has to hold for all coefficients beta,

which go along with the approximation of the test function.

Then there were some arguments, and eventually we ended up with this system of equations.