Well hello, and welcome back. We'll continue with our development of finite element methods, and we're working, as you remember, with a one-dimensional linear elliptic PDE, right? And this zeta, eta, xi model for elasticity, for heat conduction, mass diffusion and so on. So where we are is that we've written out the strong form of this PDE. We've we then introduced the weak form and importantly we demonstrated that the, the strong form and the weak form are essentially completely equivalent. Okay, that's where, as far as we got in the previous segment. And that's where we are going to continue from today. And to get us started I am just going to write out the strong form and the weak form and use that as a point of departure, okay? So. So the strong and weak forms of 1D linear elliptic. PDEs, okay? That's where we are going to start off today. Okay, so I'm going to try and write them in a somewhat unified manner, so I'm going to first write out the data, okay? And that is, given u0 we are working with Neumann conditions, remember? Just in order to not keep things too general while we're developing things right now. Okay, so given u0, t, which is going to be our Neumann boundary precondition data we're given our force in data, f as a function of x and the constitutive relation. Sigma equals Eu,x, 'kay? This is what we're given. All right, so now, let's write out the strong form. Write out the strong form here. And I'm going to use the other side of this the other half of this screen to write out the weak form. Okay, and let me also do the following, okay, do that and that, okay, all right. So the strong form is find u such that d sigma/dx plus f equals 0 in the open integral, and we have the boundary conditions, u. So, u and 0 equals u naught. And, sigma at L equals t. That is our strong form. The weak form is again, given the data find u belonging to our space S, which as you recall from our discussion, the previous segment incorporates the Dirichlet boundary condition. Okay, find u belonging to S such that for all waiting functions w belonging to another space, V, which is characterized by the requirement that all functions in V must satisfy the homogeneous Dirichlet boundary condition, okay? So find u belong to S such that for all w belonging to V, the following holds, right? And, and the important thing is that the weak form is an integral form, right? Integral 0 to L, w,x sigma dx. And in order to be able to make connections later on with multiple dimensions, I'm introducing an area of that, right? A, this is equal to integral 0 to L, wfAdx plus W at LtA, all right, so there we have it. Our strong form and weak form. Now, when we look at this, you remember also that importantly these two are completely equivalent, right? So let me write that here actually. I don't need to use all of this, so let me make this line a little shorter so I can write. I can write at the bottom of it, okay? And recall, importantly, that we demonstrated last time. Recall that the strong form. Is completely equivalent to the weak form. Okay? Now, if the strong form and weak form are completely equivalent you realize that in this sort of presentation, we have not yet done anything that may help us solve the PDE in an, in an easier manner, right? In, in, in, in essence, both these forms, the strong form and weak form are, so to speak, exact statements of the problem. And if you were to try and solve the problem in, in its exact form, there is no reason why either of these forms the strong form and the weak form, should present an easier task. A task can become easier. However, if we resort to approximations, if you choose to work with approximations of the strong form, you would be headed into something like a finite difference method where the obvious approach, would be to represent that derivative, and any other derivative that appears with the difference formulas, okay, and therefore finite differences. With finite elements we take a different approach, we will work off the weak form and introduce approximations there, okay? So the way we do this is is the following. So. Let me state here that the finite element method. Is based on an approximate version Of the weak form. Okay? And in order to understand what sort of approximations we have let me state that when we say that u belongs to s and w belongs to v. What we have in mind here are function spaces that we describe as being infinite dimensional function spaces. Okay? These are infinite-dimensional function spaces. And I'm going to tell you what that means. Okay. Infinite-dimensional is important. What this means is the following. If you are thinking of these spaces s and v as being maybe polynomials, right? We say they're infinite-dimensional in the sense that at this point, we are considering polynomials of all orders. Okay? So, you'd be considering the, constant, linears, quadratics, cubics, quartex, and so on, right? All the way up to infinity, right? What, and, and therefore infinite-dimensional, because we think of each order of polynomial as being a dimension of the space of polynomials. So let me give that to you as an example. So example, if s comma v, are. Let me put it this way. Let me use mathematical notation. So, s comma v belong to the space of polynomials of order n on x, right, where v denotes polynomials. We have in mind that n equals 0,1,2, and so on, okay. Where P denotes Pn denotes polynomials of order n. So it's in the sense that they're infinite-dimensional. Now the problem with infinite-dimensional spaces is that we're really looking for solutions in, in, in a huge space. And it is, it is the fact that we're looking for solutions in such a large space, that makes our task difficult, when we restrict ourselves to the exact statement of the problem. The idea of developing approximation methods based upon the weak form is to say that well, let's make our lives a little easier by restricting the dimensionality of the space in which we are looking for solutions. Okay? So what we're going to do here is construct approximations. In what we would call, you know, instead of infinite-dimensional, we're actually going to look at finite-dimensional spaces. Construct approximations in finite-dimensional. Function spaces. Okay, we construct approximations in finite-dimensional function spaces. All right, and example. Of course an example would be, would be to say that we construct for approximations in Pn of x, where n equals, maybe we want to say maybe we just have zero and one. Ok, and this would mean that we are considering approximations in polynomials of up to first order, right, up to linear polynomials. Okay? So this is the approach we take and, and as you can imagine now this, this is, this is likely to make our life a little easier. Because we're saying straight away that, yes I know that this the problem of interest as driven by the data that we have, is, you know, is likely to have polynomial solutions of some arbitrary order. But I'm going to try and approximate that those polynomials of arbitrary order with linear polynomials. Okay? And because it, it, because of the properties of these function spaces those approximations would be good or bad. And actually it is, it is it is addressing that question that financial element error analysis is occupied with. Okay but, we'll get to that later. All right, so, so, the way we formalize this is to do the following, okay. So, we restrict, okay, we restrict the solution space. All right, and importantly we need to restrict the space of weighting functions as well. Restrict the solutions space and weighting function space. And the solution space and the weighting function space as well. Okay, this is what we're aiming to do. And the way we write this formally is the following. We say now, now we're working with the weak form only. Okay, we say the following, we say want to find, first of all, we want to find not u itself any longer. Because we've already given up on the prospect of finding the exact solution. And u denotes the exact solution for us. Okay? So we want to say that we want to find an approximate solution, and we are going to denote the approximate solution by u sup h. That's not u, u raised to the power of h but u sup h. Okay, so you want to find u h function of x, which now belongs to a function space. But this function space is a finite dimension of function space, okay it's also an approximation. And that too is denoted by h, I'll tell you in a little bit why h. Anyway, we do the sketch. All right. So we want to find belonging to Sh. However, we want to allow Sh to lie only within the larger space of, of solutions, which also contains the exact solution. Okay? So we want Sh to be a subset of S, okay? Now, apart from that, we say that Sh consists of all function. Now we want to say something more about right? And actually, about hs itself and one of typical ways in which to do this in the context of finite elements is to say something to make a statement about the so, so to speak the regularity of our function space. Okay? I'm going to use certain notation and we'll come back and clarify this notation in a little bit. Okay? The way we'd introduce this notion of regularity is to say that belongs to a space that I am going to denote as H1 on 0, L. Okay? Apart from the fact that it belongs to the space H1, I'm going to tell you what H1 is in just a little bit. Part from the fact that belongs to H1 we do require that it satisfies the Dirichlet boundary condition. Okay? Right? So straight away, we've gone to this finite dimensional space Sh. And not only have we done that, but we've also said something more about we've, we've set ourselves up to say something more about what the space Sh is like. Okay? All right. So you want to find living in, in this finite dimensional space Sh such that. For all wh, okay? Again, the weighting functions, we are not, we are no longer pulling out of this big space V. We're taking this also from a reduced dimensional space, a finite dimensional space. Wherever we like for S, we do want Vh to be a subset of our original space V. Okay? And Vh, we will tend to say consists of all functions wh. Also belonging to the space, H1 on 0, L. Such that wh at the richly boundary Vanishes, okay? All right. So, at this point, this is purely cosmetic. Right? All all we've really done is to, is just to say that okay, we're thinking of finite dimensional spaces. I've introduced some cryptic notation by talking of H1 and I haven't yet told you what H1 is, but I will. And furthermore, I'm denoting all our approximations, all our finite dimensional functions and finite dimensional spaces by this sop H. Okay. So we're doing this, what do we want? We, we want that the weak form still be satisfied, except that now the weak form is an approximate weak form. It is a finite dimensional weak form. Okay? And it's finite dimensional, because it is computed with these finite dimensional functions. Okay? The first stone looks just as before, except that all functions are evaluated with wh, sorry, not with wh, but with the finite dimensional law versions. Okay. On the right-hand side, we have wh times f A dx plus wh at L t A. Okay? This is it. This constitutes our finite dimensional, our finite dimensional weak form. Now because in the mathematical setting this manner of introducing finite dimensional spaces is ascribed to a Russian mathematician called Galerkin. This is also called the Galerkin weak form. Okay? I'll, I will tell you in a little bit, what H is. Not right away in this segment, but in a little bit. H, H as you can ima, as you may imagine or as some of you may know bears a relation to to our finite elements when we introduced it. Okay? So we'll talk about that in a little bit. Let me see. Is there anything else I need to say about this right now? Oh, I believe not. So, so this is essentially the form in which we are this is essentially how we're going to proceed. All we've done here to see that is to observe is that when we wrote out the weak form originally the one that's completely equivalent to the strong form. We had in the mind that and we had in mind that infinite dimension of function spaces. And we recognized that can prove to be that can prove to be challenging to find solutions in, because we're looking for solutions in such large spaces. Instead, we now restrict ourselves to finite dimensional space, then saying that, that right there lies our approximation. Okay? However, we are still going to solve the weak form as, as written. Importantly as well, our finite dimensions spaces that we are, that we are setting ourselves up to use are subsets of our original larger spaces. Okay? Okay? These are subsets of our original, larger spaces. Okay. This is a good point to stop for this segment. When we come back, we will expand upon these ideas. All right?
