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Okay, let's start. So last time we discussed semi-discretization of parabolic initial boundary value problems

in the sense that we discretize in space in the way we know it already for the corresponding elliptic,

for the corresponding stationary problems, and now opposite to the linear sets of equations

which we have got as a discrete problem to be solved in the stationary case,

we now get a system of ordinary differential equations.

And as we are further on dealing with linear problems and further on applying linear schemes to linear problems,

these are linear systems of ordinary differential equations,

which can, at least in principle, explicitly, we have taken advantage of this in a theoretical consideration,

in principle, if the matrix is such that, that we know that there is and that we know an eigenvector basis of this matrix.

Which, in general, of course, to compute the eigenvalues of a matrix is a much more complicated task than solving a set of equations

or solving a sequence of set of equations.

So there's still something we should do, namely to do time discretization in addition,

which we will start with today.

But for this first intermediate step we have seen now various convergence results.

We tried a little bit heuristic approach for the finite difference case for our very specific model problem,

comparing the two representations of solutions, and we have seen, okay,

there are these higher frequent eigenfunctions in the continuous case, so to speak, up to infinity,

which do not exist in discrete case, but they are damped very strongly, at least away from zero.

That is the important thing there.

So this is an error part which is more or less negligible, and in the other, in the remaining part,

the things are comparable to the right order of convergence.

So the deviation in the eigenvalue is of the right order of convergence of h squared,

and the deviation also in the Fourier coefficient is like that,

and the eigenfunctions in this particular case even turn out to be exact at the grid.

Okay, now in the finite element case, the analysis is a little bit more satisfactory,

but what we have shown there is that we can show point-wise in time,

that is meant by this here, this situation, point-wise in time, but only in the L2 norm,

not in the norm we are used to have as the natural norm, namely the energy norm, the H1 omega norm.

We can prove something which is very similar to the behavior we have seen from the explicit representation of solution.

We have a part which reflects the initial error, the deviation between the initial data

and the Ritz projection of the continuous initial data,

damped with an exponential term with the ellipticity constant here.

We have the Ritz error, the error of the Ritz projection, so to speak,

that reflects what the corresponding stationary problem has for accuracy for each instance in time.

So you see we need already more regularity for the solution U.T.

that this thing makes sense, that we can write this thing and do these manipulations rigorously,

so we not only have to have an L2 function in time with values in H, in L2 of omega,

but we need something like a continuous function, which we also would have,

but here for further estimate of this term we need then more, and in particular we have this term here,

which is basically the same thing as here, but now for the time derivative,

also the time derivative has to be something like a continuous function in time with values in H,

and this influence is now again, it goes back over all the past, 0T,

but it's again tempted time as we have seen it for the right-hand sides of parabolic equations.

And the background of this proof, and maybe I come back to that because it was maybe a little bit confusing what I said,

so what we do is we have to estimate the error here.

We subdivide the error into two parts, namely the deviation between our, say, finite element solution

in the semi-discrete case and the Ritz projection, so this is something which takes values in VH,

this is called theta, and then we have an additional term which is just the error of the Ritz projection, which appears here.

So if we are able to deal with the first term, and if we deal with the first term leads to the first and the third term in the estimate,