mis receiving

a unique solution for the heat equation.

And the heat equation is only a paradigm.

And in the second step, we will then see

how to extend this proof to more general cases.

And what we have seen up till now,

our approach is the implicit Euler method.

And what we have seen meanwhile is that

from the implicit Euler method,

we can construct, of course there are other possibilities

too, but two are obvious,

we can construct two time-space functions.

So we have, so to speak,

via implicit Euler,

we have for a sequence in time,

we have functions or elements of the Banach spaces,

which we denote by u,k,

from h one, zero of omega,

which is our basic space.

And from this we can construct two functions,

the one we call u tilde n,

n is the parameter of the discretization,

so we have n plus one points in space.

So this is piecewise constant in time,

and we have also defined a name for this space,

namely we call this then the vn functions, I think.

So this is element vn,

vn meaning those functions which are, let me write it a bit,

shorter, if I abbreviate this with x, is l to zero tx.

Let's do it like this, piecewise constant,

in time on this fixed discretization of time.

So that is one version,

and the other version is without the tilde,

these are the piecewise,

these are the affine linear continuous.

So if you want, these are the linear splines,

the S1 splines, these are S0 splines,

with Banach space valued functions here,

and this space we correspondingly denote with wn.

Okay, so what is now the problem to go to the limit?

So now we have seen by doing energy testing

with the solution itself,

we have seen estimates, namely,

namely, where are we?

We have seen that we have,

yeah, first of all, let's write down the equation.

So if we want, we have now the following equation,

time derivative of un,

because this is just a difference quotient on each interval,

that is what, in our definition of the implicit Euler,