All right, welcome back. We continue with developing our finite element method for linear elliptic PDEs in three dimensions, with scalar variables. Okay? So, let's let's just start by recalling the setting, that we have. So, remember this is where we are. We have our basis vectors, we have our domain of interest, over which we are solving this problem. And, if we are done thinking of heat, heat conduction or mass diffusion again, we call that we have two different parts of the boundary. I think maybe I call the maze the Dirichlet boundary. This could be the Neumann boundary. Over the Dirichlet boundary, we're controlling the field itself. Either the temperature or the concentration field for the diffusion problem. Over the Neumann boundary, we are control, controlling the influx, right? The influx of heat or mass. All right, and we have distributed sources. All right. For, for, for either physical interpretation of the problem. Okay. So in the last few segments, we started out by looking at, we started out with the strong form of the problem. And then derive to weak form. I, I stated that they are completely equivalent. I showed the equivalent in one direction. From strong form to weak form. Okay? And we spent some time understanding all the different terms in there. So this is where we are. Where we were, at least. And what we're going to do today is pick up from that weak form, and take the steps that will eventually get us to our finite element equations for this problem, right. And as you will recall from the 1d problem what that means is that we need to work out the finite dimensional weak form. All right? And, for setting the context, let me go to our usual sketch. We have our bases. And that is our domain of interest, omega. We have some point in there which has position vector x. I'm not going to draw the position vector just not to get this diagram too busy, and we have here our boundary conditions. Bound different boundaries, right. We may have, we have the Dirichlet boundary, and the Neumann boundary. Okay? So this is the setting that we have and the, the infinite dimensional weak form that we have already derived is the following, right? I'm, I'm going to first write the infinite dimensional weak form, and then we get to the finite dimensional weak form, right? So what we've got as far as, is the following. We've said all right, let's what we need to do here is find u belonging to S, which includes the Dirichlet boundary condition. Right given all the usual data in the problem. Given, sorry I called this u not, it's ug. Given ug. Given our mass influx. Given our forcing function, and given our constitutive relation which is j equals sorry, I'm going to, I'm going to stick with writing this in coordinate notation. So I'm going to write this as ji equals minus kappa ij, u comma j, right. Given all this, find u such that, for all w belonging to V, which consists of w that vanishes on the Dirichlet boundary. Okay? For all w belonging to the, to the space V, we have to follow in weak form. All right? We have integral over omega. W comma i. Ji, dv equals integral over omega w, f, dV minus integral over the Neumann boundary. W, jn, dS. Okay. Let me just make sure that this all works out, yeah. All works out, right. So so, so this is our weak form that we derived. And of course, you know, at this point, we haven't said anything special about our spaces S and V, except for the fact that S includes the Dirichlet boundary condition, v includes the homogeneous Dirichlet boundary condition. So at this point, when posed as such, we are really talking of an infinite dimensional weak form. And as we made the observation in the case of the 1d problem that does not make it any easier to solve than the strong form. It bears complete equivalence to the strong form. And so we really haven't made any steps towards making it easier for us to solve or to deve, towards developing approximation. Right? And just as before, we develop approximations by going to a finite dimensional form. Right? So the finite dimensional form is the following. Right? Just the statement of it is going to look very much the same as before for the 1d problem, right? So what we want to do now is find belonging to Sh, right? Which is, is a subset of S. Okay? And as before Sh is a finite dimensional function space. All right, okay let's say a little more about Sh, all right. The way we construct Sh is again going to look like we did like, like everything we did in the 1d problem. Sh of now consists of functions like. Right. And now we specify as we did in the 1d problem that we are interested in functions that live in each one. Right, over the domain of interest over omega. All right, and because Sh is a subset of S, it must inherit the Dirichlet boundary condition as well. All right. So equals ug on the Dirichlet boundary. Okay. All right. And then the rest of it just follows right, given everything else that we have. Of course, we've given ug. We are given the influx condition. We are given the forcing function. And we know that the same constitutive relation applies. Right? Okay. I guess properly when we are stating the constitutive relation, the context of the finite dimensional weak form, we are no longer speaking of u being drawn from the full space S. So, we can already put an h there and there. Okay. And you recall that just as we did in the 1d problem, the h is the soup. H is just to remind us that these are finite dimensional functions and the way we construct them of course, critically uses the, the notion of an element size. Okay? So that's what the h show, indicates. All right. So, we're going to find in this sort of space. such that For all wh in vh subset of v, and where vh consists of functions wh. Also, in each one. On an eh, over omega. But satisfying the homogenous Dirichlet boundary condition. Okay? So, for all such wh again, the finite dimensional version of our weak form is satisfied. Right, and that takes on the form, integral over omega. Wh comma x, sorry. I'm lapsing to my notation for for the 1d problem. Wh comma i, jhi, dV equals integral over omega wh, f, dV minus integral over the boundary, wh, jn, dS. Okay, and once again, we observe that the data are not finite dimensional function. Right? We have exact representation of that data. Right? Okay. And, and you'll note that really in writing out this weak form, the only thing we did was replace all our any function that is drawn or, or obtained from u or w with the corresponding finite dimensional version. And of course, we defined what the finite dimensional spaces are going to be, right? As for the 1d problem, we are drawing then from H1. Okay, so this is our finite dimensional weak form, for the problem, okay? And, and you recall now that as in the 1d problem, what we need to do in order to proceed with the, with the formulation is to define what we mean by these finite dimensional spaces. Okay?