The following content has been provided by the University of Erlangen-Nürnberg.

Last time we started with discussing the condition number of the stiffness matrix.

That is now the aim to get an estimate for the condition number of the stiffness matrix,

which is denoted by A.

It will be only an estimate from above, so one can argue maybe this is not sharp what comes out,

but in fact we will see with an example later on that this is in fact a sharp estimate.

And as an auxiliary tool, later on it will become a matrix, so to speak, of its own right.

We consider the mass matrix.

It's similarly built as the stiffness matrix, but we do not take the bilinear form of the variational formulation,

but we just take the L2 scalar product to form the components of the matrix B.

So the B always is a symmetric matrix, symmetric positive definite.

Nothing can go wrong with this matrix. With a matrix A, of course, it depends on the bilinear form small a.

The most simple situation is small a, and with that also capital A is also symmetric,

and we will help us a little bit by using an assumption which basically means that we do not deviate so much from the situation.

So that's maybe not 100% satisfactory.

So as auxiliary results, we have investigated the following discrete L2 norm,

which is composed on our ansatz space by using the corresponding triangulation,

the corresponding degrees of freedom situated at the ais, and we take the squares of the values there,

sum over the elements, multiply with dk to the d, sum over all the elements, and take the square root.

So it's quite similar to what we did in the beginning discussing the finite difference method,

where we had a similar norm which we call discrete L2 norm, but in this norm,

and in some sense the weight with the d, forget about the dk for a second, and take the hk for a second,

and take hk as h, or more precisely later on we assume quasi-uniformity,

that we have above, above from below and above between hk and h, the maximum of the hks,

then we would have here the same weighting factor as we had in two dimensions, h to the d halves,

in two dimensions this means h, and that was the weighting factor we used.

But what is different here now is that depending on the triangulation,

we so to speak collect these values squared several times,

because a node belongs to several elements, and without knowing something about the triangulation,

in the general case we don't know how often it disappears.

We have introduced this semi-norm because what we can, so what are the general assumptions we are doing,

we are assuming, where is it, that the family of triangulations is regular,

that we have an estimate between the rho case and the h case,

and the additional assumptions are basically the ones we always do,

so we have a continuous and V-elliptic form on a subspace of H1 omega as usual.

So this is a norm, this is just the only question is the definiteness and the uniqueness of the interpolation,

and it's a norm of interest because in fact it's a norm which is equivalent uniformly in H,

that is what is stated here, to the L2 norm.

Of course, if I would fix the phi H, then of course every norm is equivalent to every norm,

that is not an interesting statement.

I have to know what happens if H goes to zero, and in this sense I have here uniform equivalency,

so this is actually L2 norm with which I am working,

and on the basis of that we have seen this inverse inequality saying that what is of course wrong

on a full finite dimension, infinite dimension space that we can estimate the one norm by the zero norm,

we can do here, but to the expense of the finite dimension space, but to the expense of one over H.

Okay, the proof was quite as we are used to do them,

so we are dealing with integrals, we subdivide the integrals and integrals over the elements,

which transform the elements to the reference element and so on and so on,

somewhere we use Bramble Hilbert or something like that.

Okay, so now two statements are left to be chosen and maybe you hopefully have forgot the last two sentences