Welcome back. We'll continue with our development of the finite element method for these linear parabolic problems in three dimensions where we have a single scalar unknown. What we've managed to do in the last couple of segments is starting from the weak form we've written out the stro, sorry, starting from the strong form we've written out the weak form and arrived as far as the matrix vector equations, okay? And if you have your notes in front of you, you can go back and observe that the form of these final matrix vector equations was the following. Right, the matrix vector equations. Right, it takes on the following form. We have this new matrix, the mass matrix Md dot plus Kd equals F. Now in this form of the equations, we know that because of the, there is no methods being employed. The methods are on the weak form. The the boundary conditions are already embedded in this set of matrix equations, right? What we do need to count for, however, is the, the initial condition. Okay, so whereas we have this linear system of equations, right, essentially an a, a first order of ODE, right? So this is a first order ODE. ODE being ordinary differential equation, so let me write that out. Right, in our vector unknown, d, okay? Now, because we've dealt with all this business about, you know, number of nodes in the problem and the number of Dirichlet boundary conditions and so on, I'm going to, as we proceed from here on just say that d is a vector living in an ns ndf dimensional space, right? NDF is simply number of degrees of freedom, okay? So, those are the number of degrees of freedom we are served. NDF is the number of degrees of freedom we are actually solving for. Now, this is the first order ODE, and what that implies is that we do need an initial condition which is something we had stated in the strong form of the problem. We just need to incorporate it here, and that ODE that we're working with i, is the following, sorry, that dif, i, initial condition that we need to work with is the following. d at time to equals 0 equals the vector of conditions which are obtained by taking u not at the particular location of the corresponding node, okay? And this is all the way down to u naught at x, now written in ndf, okay? Right, instead of continuing to number by node, we've forgotten all about nodes. Now we're just looking at this linear system of equations where d as an R belongs to to an ndf lucrative space and this is what we have for the initial condition, right? It's convenient sometimes to also offer, to, to refer to this vector as just d naught, okay? Right, so this is the setting we have. One thing I should mention is that whereas we sort of carried through our development of the method by considering a consistent mass matrix, one can also do global lumping, okay? So a globally lumped matrix, lumped mass matrix. All right, let's denote it as M sub l for lumped, okay? It can be defined as this thing can be defined. Okay? And the way to do that is to simply say that MAB lumped is equal to the sum over c of MAC, okay?. If A equals B, right, and it is equal to 0, otherwise. All right? Okay, and in some cases the problem is often solved with the globally lumped mass matrix. Okay, so this is this is what we needed to complete from the previous from, from the work, we set, set out in the previous segment. We move on from here and begin looking at the integration algorithms for this class of problems, okay? And this is going to depend upon time discretization, okay? The reason for it is that the the form of the matrix vector equation we're working with is what we call a semi-discrete formulation. And I explained why it is semi-discrete because we've gone through the discretization in space, but don't get in time. Okay, so now we do time discretization. Now, in a move that may seem a little disappointing to some of you the time discretization for this sort of problem is done with finite difference methods, okay? However, the, the nature, the structure of the matrices M, K, and and the vector F depend upon the finite element method. So even though we're going to use finite difference methods in this series of lectures for integration of time dependent equations, time integration of time dependent equations. Those methods are, do continue to be affected by the, by the underlying finite element formulation for the spacial part of b d e, okay. So this part is going to be finite difference. Finite Difference methods, 'kay? With that, that's what we're going to use. Now, I'll make a remark here, which is that a fully, space-time, Galerkin finite element method does exist. Okay. So, space-time. Finite element methods do exist. And they have remarkably good properties as well. These types of methods are based on, carrying out integration. Over, omega and over the interval 0, T. Okay? So these are based actually on a formulation which is now going to look, well, I, I, I, I probably shouldn't since, begin writing this formulation because one, we'll get sucked into the rest of the development, but anyway, they're based upon integrating over omega and over, the time integral. And, they, they tend to have very good properties. For instance, they have properties of the type where, the accuracy. The accuracy with respect to time. Is of higher order. Then what we will be developing here. Right? Accuracy is of higher order than, with finite difference methods, F D for finite difference. They do come with a, they do come at a cost, though. In particular, one has to develop a finite element mesh over space and time. And this can pose a challenge when one is doing problems that are 3D in space, right. Because then you have a forth dimension in time, which can get to be a bit of a challenge to visualize, definitely. At any rate the methods exist and they're actually very sophisticated, okay. So we get back now to our finite difference methods, right. So the nature of the time discretization that we are, we are seeking carry out is the following. We are going to divide our time interval into sub-intervals. So divide our time integral 0,T into sub-integrals. Okay. And these sub-intervals are, 0, let me do it this way. These sub-integrals are t0 to t1, t1 to t2. So on until we come finally to. TN minus 1, to t N, okay. So we have all of these sub-intervals, and what we also imply here, of course. Is that t0, or t nought is equal to 0, and tN equals capital T. Okay. So, if you look at this what we've implied is that we've divided our total time interval into capital N sub intervals, right. So clearly here we have, N sub-intervals. Okay? The basis of our, of the methods that we will develop is the following, okay. So what we are going to do here is consider. A, consider an interval. T sub little n, to t sub n plus 1. Okay? All right. Where, where, of course, little n belongs to the integer integral 1, 2, N minus 1. Sorry, 0 to N minus 1. Okay. So we consider an integral of this type. The whole, basis of our methods is going to be what is sometimes called time stepping, okay. So, time stepping implies. Right. Time stepping implies that we know the solution at tn, and we want to get to tn plus 1, okay. Time stepping is essentially, is the following. Knowing, the solution at t sub n, find it at tn plus 1. Okay. All right, so we're really, sort of, hopping along in time. Okay, so if you think about it as, if you think about it as follows. That's the time axis, right. We've started here at 0, we want to get as far as t, right. We have t0, t1, right. We have tn, and we will have tn, t little n plus 1, and here, of course, we have t capital N, okay. So the whole idea is that well, I know what, what, I know everything at tn. How do I get to tn plus 1? Okay, in order to proceed, and also actually in order to Later on carry out some analysis, we need to lay down some more notation. Okay? And here is what we will do. So some more notation. Okay? When I write d, at some time t n. What I implied by this notation is the. Solution vector d, if one were able to do exact integration in time. Okay. So this is the time exact solution, or the time continuous solutions. All right. At time at t equals t n. I am going to stick with the term time exactly. Time continuous gets, gets a little bit more, ambiguous. The time exact solution. Okay? And this simply means if we were able to integrate our ODE exactly. Right? This is our matrix of vector ODE. Right? If we were this with, with bound, with initial conditions. If we add a method to exactly integrate this, then the solution that we would get at a given time t n would be this, okay. In contrast, I will use d sub n, okay, to denote the, the algorithmic, solution at t n. Okay? So this is the algorithmic. Solution, okay? By which mean the solution that's obtained by applying some sort of discretized time integration algorithm based upon the discretization that we've just spoken of. Right? The algorithmic solution obtained. By the method to discrete, to integrate the discretized, the, the time discretized ODE, okay. All right, so we have this notation. What we need to do is actually first of all say what we mean by the time discretized ODE. And here's what we mean, okay? Now if I were to simply re write out our ODE in the time-exact form, here's what I, I, I would write it as m just as I have it up there, but let me, explicitly put in the time. Okay. Supposing we were looking at it time t n. Okay? This would be m d dot at time t n plus k d at time t n equals f which potentially could also depend upon time. All right. Now when go to a time discretized version of this ODE, what we will write here is the following. We will write this out in the canonical form m. Now instead of d dot t n, we will adopt the notation that, you know, that well if d is our solution, we can think of d dot as sort of the velocity of the solution, right. Just, just as a term, it's really just the rate of the solution, but, but we use the, the sort of canonical or the generalized velocity for a rate here. And so we would write this as v, okay? Indicating a velocity. Now, this is not a time-exact quantity. So we are going to denote it as v sub n, okay? Using this sort of idea. Okay? Plus k d n equals f at n. Okay. This is what we mean by saying that well here, v at n is an, a discretize approximation, It's a time discretized approximation again. Time discretized, discretized approximation of d dot right. D n is of course just as we've defined above. Okay, so this is what we call what I will refer to as the discretized version of the ODE, okay. So this is the time discretized Are the time discrete, ODE. All right? Okay.
