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Okay, last time we have introduced the five-point stencil as the most simple stencil to discretize

the most simple elliptic PDE, namely the Poisson equation with Dirichlet boundary conditions.

And today we want to start to analyze what we did.

So now we have sort of a feeling how to do it, but we have no idea what it's good for.

To fix our notation a little bit, I want to point out one thing which I have forgot last time.

That is the notion of grid functions.

So I said already, or we have seen this already, that one of the specific things of finite difference schemes

is that we have the grid points and that is where, so to speak, everything lives.

The domain in between is more or less non-existing for the method,

which is also one of the drawbacks of the whole approach.

So if we have values at this set of grids, then this defines a function, of course.

It's just a mapping, we have a grid point, and we have a value at the grid point,

but it's difficult to speak in a reasonable sense of properties of this function.

So we have no continuity in a reasonable sense, no differentiability and so on.

This is what a grid function is.

It's just a mapping from the set of grid points to the reals.

Either the grid points or the closure of the grid, the grid is either omega n, omega bar, or the boundary depending on what you would like to have.

On the other hand, of course, we have the representation of a grid function by a vector.

But for getting a representation we have to choose a numbering.

We discussed different numberings already yesterday, the last time, and we have the impression that the choice of the numbering

should not have too much influence on the method itself, on its properties.

It has influence on the structure of the matrix which comes out, which may be of advantage for the methods chosen to solve it or not.

So we distinguish a little bit between the grid function UH, which is written non-bold, and the corresponding vector UH,

which is the written bold, which comes out after selecting an order.

And the same applies for the data.

So here, for example, we've chosen the order and therefore in bold print we have a UH.

And the same applies for the data which go in.

So there are the given continuous functions, the F for the right hand side, the G for the boundary conditions,

we have the evaluation at the grid points, this gives us grid functions, and then we have vectors representing the grid functions.

So this would then lead, in this representation we get the bold F as the vector representation of the grid function FH,

and the G as the vector representation of the corresponding grid function.

So also I've omitted here the H, not to overdo the notation, it's something which always depends on the H,

because with every new H we have a new grid, and we have a new evaluation of these functions.

And also we are only dealing here in finite dimensionless spaces,

so we have a finite M1 as the number of grid points in the interior, where our unknowns, that's the number of our unknowns,

and a finite number of M2, the amount of typically boundary conditions,

these are not in the process of convergence we are now going to look at, and I mean the process, what happens if H goes to 0?

These are not fixed numbers, they go to infinity if H goes to 0.

And that is the whole thing we have to deal with. We are not dealing with one specific finite dimensional problem,

we are dealing with a whole family of finite dimensional problems for all ages greater than 0, or in a certain range, in any case going to 0.

Okay, so this is what we have all discussed, and now let's see what kind of questions we want to ask.

At the end of the day what we want to have is we want to have convergence.

And as our method only produces values at the grid points, grid functions,

the only thing which we can naturally do is to compare the solution, the solution which hopefully exists,

which of course we generally do not know explicitly, that's the reason why we do numerics,

we can compare the solution with a computed grid function and say something about whether this is a good approximation or not.

So we produce a grid function, capital U, out of the exact solution small u,

and this again gives then rise to a vector representation which is in a small bold U.

So what we want to do is we want to compare these quantities.