What we are going to do, is to use this as background to define to define norms. Okay? We are going to restrict ourselves to defining a, the sort, only the sort of norm we, we, we want, okay? So what we will do is define the H1 norm, okay to be the following, okay? So supposing we're taking a func, we have a function, v, okay. We're going to define the H1 norm of v, and we will write it as that. Okay? The double lines are, are a kind of bracket. They're they're usually used to designate a norm, and that's subscript 1 indicates that we're talking to H1 norm. This is defined as the following 1 over measure of omega to the power of 1 over number of spatial dimensions. Integral over omega, v squared plus measure of omega to the power 2 over number of spatial dimensions, 'kay? Of v comma x squared dx. All of this raised to 1 over 2. Okay. Just a couple of things to notice here. We have divided through by measure of omega raised to 1 over number of spatial dimensions to get rid of the effect of having integrated, the function and its derivative squared over the domain. Okay? Defined as this, we have what is called the H1 Hilbert norm of our function v. Okay? Hilbert norms are, are a special case of a more general type of norm that's called a Sobolev norm. Okay. This is a this is an example of a, a, this is an example of more general norms called Sobolev norms. Okay? The idea of Hilbert and Sobolev norms come from functional analysis in Hilbert and Sobolev spaces. The fact that we are raising the function to the 2nd power as well as its derivative to the 2nd power, and then taking the one-half power of the whole thing is what makes it from a Sobolev norm in general to a Hilbert norm. Okay. 'Kay, for Sobolev norms, you would be raising to other powers, and also sort of taking that, the root of that corresponding power, right? So that, that, that leads to Sobolev spaces. Okay, so, but, but this is what we need to work with here, and so we work just with just H1 Hilbert norms, okay? Or we will just recorded them as H1 norms. Now note that you get a, a special case. Not, this indeed is actually a special, special case of a more general way of defining even a Hilbert norm as one can extend this to, to the Hn Hilbert norm. Okay? By, by just taking up to n derivatives instead of a single derivative as, as I have done here. Okay? So we can extend this to define the Hn norm, 'kay? By including the first n derivatives. Okay, instead of taking just one derivative here. Okay, now, the fact that we can do this also allows us to define what is called the H, what we may choose to call the H0 norm. Okay, by just saying well, I don't, I don't consider any derivatives, I just consider the function itself. Okay? So the H0 norm would then be denoted by either 0 here or it's more common not to put anything. Okay, so when I just write this without any, without, without the subscript 1 as I have in the first equation of the slide, we have what is called the H0 norm. Okay, which is just now 1 over measure of omega 1 over nsd integral over omega v squared dx to the power 1 over 2. Okay. This is called the H0 norm. Now, you note that the way we first talked about H1 functions. At that time, we also talked about another class of functions called L2 functions, which were simply squared in the. Right? Okay? And, and in that case, we just simply multiplied the function, we squared the function, and integrated over the domain. Right? And, we said that if that, this quantity in fact is integral that we've written on the right-hand side, we said that if it were bounded, the function is an L2 function. Okay so this is entirely equivalent to also the L2 norm. Okay. So in fact what we have here on the, at the bottom of the slide is more commonly referred to as the L2 norm rather than the H1 norm. Okay, that is more common terminology. But, but you see that the L2 norm is basically the H0 norm. Okay, so this is all background. What do we care? Where does any of this help us? Here is a result I'm going to put down that we will use today. Okay. The result is the following. We can also sort of leading up from this idea define what we will call the energy norm. Okay. You can also define the energy norm of v, okay, which is the following. We will denote the, the energy norm as I'll tell you what it is first. It is integral over omega, okay, of v comma x, E v comma x, dx. Okay? All of this to the power one-half. Okay? This is the energy norm of v. Why is it called the energy norm? This is a throwback to the time when structural mechanics was considered problem that finite element methods were applied to. In that setting, if v were your, your, if v were your displacement field, then the quantity I've written out here will be related to the strain energy, okay. So this comes from the notion of the strain energy of v. Okay, you would tend to think of v as a modulus, and therefore, you would conclude that this is a strain energy. Okay? So this is the energy norm. Okay? All right. We have a result which is a result on equivalence, equivalence of norms. Okay? And that result is the following. Okay? It says that if we take v, we form it H1 norm, and multiplied by some constant, say c1. Okay? We're seeing that this H1 norm of v, bounds from below, the energy norm of v. Okay? Okay? It can be a lower bound of the energy norm of v, which itself can be an upper bound of the H1 norm, sorry, which, which itself can, can be another load bound also of, of the H1 norm. Okay? Alternatively, what the statement says is that essentially what this thing is saying is that we have the H1 norm of v and all we have in these two terms that I put arrows on. The only difference between them is the fact that they have different constants. Okay? What we're saying is that just by multiplying it by different constants, it is possible to bound the energy norm of v from below and above. Okay? All right. So, the only way this is possible is if, fundamentally the H1 norm and the energy norm behave in the same manner. When one goes up, the other goes up too. When one goes down, the other goes down too. Okay, if that is the case, then we can always find these constants c1 and c2, such that you can bound the energy norm from below, and below like here, or from above like there, using the H1 norm. Okay. So essentially this, what this thing is saying is that one can bound the energy norm from below and above by the H1 norm. Okay, that really the only difference is a constant of multiplication, okay? So it's in this sense that we, that we say that the energy norm and, and this H1 norm are, are fundamentally, you know, they're, they're fundamentally equivalent. Okay. To end the segment, I just want to pre, present a little more notation, okay. It is the following. We will define what we will call inner product notation. Okay? So. We will define w comma f as written like that to be the integral over omega of w times f dx. Okay? Right. We will define as well the following. We will see that, so this is set to be the inner product of w and f. Okay? In particular, this is what is called the L2 inner product. Okay, and you just take two functions, multiply them together, and integrate them over the domain, over which both functions are defined. We get the L2 in our product. Okay? We also have what we will call a bilinear form notation. Okay? And for our focuses, we will define the bilinear form of w and u. As this to be the integral over omega of w comma x, E u comma x dx, okay? All right. You observed where this is coming from. Where does the integral on the right come from? Why are we defining it in this form? What is motivating it? Right, what is motivating it, is that is the fact that this integral on the right is, in fact, the integral that shows up in our weak form, right, this is what leads to the stiffness matrix. Okay, and in fact the inner product that I defined up here is exactly the sort of inner product that shows up on the right-hand side, right, the forcing function. Okay? All right, so this is motivation for defining them. Why do you think this form is said to be bilinear? Okay? Right. The reason it's bilinear is that it is linear in w, in the function w. As well as in the function, u. Okay? So it is bilinear in w and u. Right? Or, or alternatively, there's linear in each of w and u. Okay. Therefore, we say this case is bilinear. Okay, so that is perhaps the bi here is superfluous but I use it just to re-emphasize what's going on. It's individually linear in w and u. 'Kay? Last question for this segment. Do you see a relation between what I've just written between this bilinear form and the energy norm, that we defined in the previous slide? Right. Note that a of u comma u, right? If we use the same function in both slots of the bilinear form, we essentially recover our energy norm. Okay. And what comes ahead of us, the analysis, we will tend to use this notation, a of u comma u, for our energy norm of u. Okay? Okay, we'll end this segment here.
