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Okay, let's start. Before we now begin with the finite element method, which will keep

our attention for a long time, I will make a few remarks still to the finite difference

approach. Maybe we will come back to it a little bit later, but only after we really

have dealt with the finite element method extensively. So the one remark concerns a

little bit of a refined argument compared to what we have done already. So let's call

this a refinement of the order of convergence estimate. Always in the framework we have

discussed last time, in particular meaning that we always assume that we are dealing

with discretizations where the discretization matrix is inverse monotone. And we have seen

requirements which are sufficient for that. And you remember the basic thing was to also

in this, under this assumptions also to come to stability, one decisive step was to look

at grid functions WH such that AHWH is greater or equal in the point-wise sense than the

vector one. On the basis of this, having such a function we have seen that basically on

the basis of the infinity norm of this function we could build up a stability estimate of

course in the situation where we have a uniform bound on this L infinity norm. There is a

little bit more in this approach. We can do the estimate a little bit more refined. And

this is then also the basis why we can do a little bit better in situation where we

up to now thought for example in the situation on general domains with then non-equidistant

stencils. We up to now we said okay if this is necessary then we only have first order

of convergence because near to the boundary we have a stencil which has only first order

of convergence. This does not need to be the real truth as we will see. So what can we

do in such a situation a little bit differently to what we had till now. So if we look at

the grid function U, so this is just the evaluation of the exact solution, if we plug it in then

we get so to speak the normal right hand side and we get something which I now call QH hat

which is just the consistency error. That is just the definition of the consistency

error. We have our consistency error estimate. So we also so to speak the further assumption

is that we know the consistency error in the discrete L infinity norm behaves like a constant

2 times H to some power of alpha. That was always the general situation we had. So now

let's modify our argument a little bit by looking at a function W tilde where we just

take this W but scale it. Let's call this C1. So I had it also C1 here. Scale it with

this factor here. So what does that mean? Then just by plugging it in if I again define

with EH now the error, so EH should be UH, the grid function we are computing which our

approximation minus the grid function U, the exact solution. But this is now a grid function.

It's not a norm. We look at the full error function, not at a norm of the error function

at the moment. So what do we know? We know by this definition and just by this estimate

here we have on the consistency error that if I compare AH applied to E epsilon, this

is error function, and I compare AH applied to the W tilde, I have less or equal. Just

put in all these inequalities. Then you see this immediately. But by inverse monotonicity

this means that we have an estimate which is now not a norm estimate but a point-wise

estimate at each grid point. A point-wise estimate says that the error grid function

can be estimated by C1H alpha times omega H. Analogously we can do the same estimate

from below saying that we get then minus C1H alpha omega H less or equal to EH or put it

together we have, and this vector is now a vector still meant to be the vector composed

of the moduli of the components, can be estimated by C1H alpha H to the alpha omega H. So now

of course I can go on with this estimate and say of course I can estimate this now from

above here with the say infinity norm of this vector, and if this is again stability bounded

by some let's call this I think C bar, then we are in the L infinity norm estimate as

we are used to. But the step in between this has a little bit more. This is now a point-wise

estimate here and of course the shape of this function W now enters into this estimate.

And if you remember the functions W we have constructed last time, this were these parabolas