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First I want to say, because it's not so optimal here in this room and in the neighbouring

room also for taking the video, we've now got, at least for one of the days, for Friday

we got a lecture on Friday. On this Friday on we will be in the Part 12, page 12 over

there. Okay, so let's start. We developed a finite element method for the specific case

of the Poisson equation with homogeneity and sterically boundary conditions. And we saw

in general what the Galerkin ansatz means, namely solving the variational equation not

on the full function space but only on a finite dimensional space, means to solve a set of

linear equations. In general that means independent of the underlying problem of the underlying

bilinear form and linear form on the right hand side, independent of the basic space

V and independent of the ansatz space VH. In all those cases if we choose a basis phi

1 to phi m of the ansatz space we see that due to the bilinear nature of the form here

the requirement that we are looking for a solution UH from VH such that the variation

in the quality of what is true for all test functions from VH is equivalent to this set

of equations where the system matrix, the so-called stiffness matrix comes to existence

by inserting the basis functions in the bilinear form, the ijth element is a phi j phi i, so

just the other way around and the right hand side is just the evaluation of the functional

of the right hand side of the basis function phi i. So, now clear what the basic things

to do is, first is the assembly phase that means to set up the set of equations and the

second is of course to solve the set of equations. And on the other hand if we do exactly the

same thing for the Ritz approach where we know already that not in the general case,

this was a very general consideration for example it did not need the symmetry of the

bilinear form. The Galerkin approach also makes sense if the bilinear form is not symmetric.

That is not the case for the Ritz approach but if we have this symmetry in addition then

we know these two approaches are equivalent and positiveness we also need, not really

definiteness but positiveness of the bilinear form and if we, they are equivalent on V but

they are equally equivalent on VH, they are equivalent on any space where they are defined

of course also then on a finite dimensional one. And that means we have the same thing

here in the finite dimensional setting if you work out what this finite dimensional minimization

problem is, it turns out by exactly the same consideration that it gets this quadratic

form again with this already defined matrix AH as the matrix defining the quadratic part

and the vector QH in defining the linear part. So what we see here again is the equivalency

of the set of equations and of the minimization problems that is of course not true for any

set of equations but you might know that it is true for symmetric positive definite matrices

and that is also the background here. So if we now look a little bit at the properties

of this matrix AH we can see several properties. The first thing what we see if the underlying

bilinear form is symmetric, that is the case in our example, that might not be the case

in more general examples if convection is present and is not the case anymore then also

the matrix is symmetric because the matrix is just A phi J phi I. So if A is symmetric

of course the matrix is symmetric. The other thing is if the form is positive definite,

then if the form is definite then also the matrix is positive definite. So this is just

this framework where we have the equivalency, basically the framework with the equivalency

of the two formulations Ritz and Galerkin and therefore we also have then the equivalency

of set of equations to solve and minimization problem to solve. Why is this matrix positive

definite? So we just reverse all our calculations what we did. So we take some vectors, some

tuple vector xi here and correspondingly we have the element of the function space which

is the linear combinations out of these coefficients and the basis vectors. We just have some fixed

basis, any basis in a general situation. This is a consideration general situation. So we

write down what this quadratic part of the Bialyner form means. This is this thing here

and now we just reverse our computations. Now we bring in the coefficients and we bring