The original Poisson equation or the wave equation for that matter, is actually called a strong form of the equation. What we will now develop in the next few minutes is actually called a weak form of the equation. So, there are strong forms and weak forms, and eventually, we're going to solve the weak form of first, the Poisson equation and later the elastic wave equation. Now, what does the weak mean? The weak actually means the solution is no longer absolutely accurate, but it's actually only valid within the space of so-called test or basis functions. So, how do we modify the equation to get to this so-called weak form? Well, the first thing is we multiply the equation on both sides with an arbitrary or not known but mathematically well-behaved test function and we'll call it v. V is a function of x and it's defined on the entire domain. The next step is actually we integrate on both sides, again, over the whole physical domain which we denote by D, so we have on both sides now an integral. We can do this, we have not changed the solution to that equation, that's just a mathematical modification that we can do. We now see in the next step what this actually leads to. The first step we're going to do, is actually doing an integration by parts and the general form is written here. We have an integral over g and f prime, prime here is the actual space derivative. Integral over the whole physical domain, in this case, and it's leading to an anti-derivative and another integral containing derivatives and the original functions. So, we're going to apply this to the left-hand side of the Poisson equation that contains an integral over the shear modulus multiplying the second space derivative of u, times the test function. We're going to see what that leads to. So, let's do that. When I make the following mapping, f is the first derivative of u with respect to space, and g corresponds to our test function v. Now, what happens if we apply integration by parts to this left integral in the Poisson equation. Well, we end up with an antiderivative that contains the first derivative of u with respect to x, and another integral that actually contains Mu and the space derivative of u or displacement field and the space derivative of our test function v. Now, something important happens. Actually, we can either assume that u, the field that we seek is zero at both boundaries, then this antiderivative would be zero or that the space derivative of u with respect to x would be zero, so the gradient will be zero at those boundaries. Actually, the latter exactly corresponds to the so-called free surface boundary condition that at least in seismology the stress free condition at the boundaries that we often have to fulfill. That's a very interesting point and an important situation for elastic wave propagation when you have a free surface boundary, because actually, that boundary condition comes for free. It's implicitly solved and that has really very important positive consequences for the simulation of such situations. Actually now, if we turn zero, if we allow this term to be zero, the antiderivative we're left with this integral that we put on the, again, replaced the left-hand side with this, now with the minus sign is gone, it's a positive integral and actually that is called the weak form of the Poisson equation, and this is the equation that we're going to solve numerically. At the moment, this is still defined in the continuous world and we now have to actually make the step and discretize that equation. That's in the next video. So, how to get into the discrete world. Well, actually I would do something that we've seen before when we talked about function interpolation. Let us replace our exact u by an approximation that we actually call u bar. This u bar is equal to a sum over a set of basis functions that we call Phi i, we don't specify them yet. They're weighted by coefficients that we actually also called u subscript i. Now, with this approximation, we go into the equation again, the Poisson's equation, because of course we require that our approximate function also solves that equation. So, we end up with the same integral equations on the left hand side. We'll take Mu. We now assume that Mu is independent of x, so it's homogeneous, a homogeneous medium so we take it out of the integral. So, inside the integral, we now have our approximation u bar which is now a sum over some weighted basis functions. Now comes the most important step in basically all finite element type analysis. As test functions, we're also using the same basis functions Phi now j, Phi subscript j that we've actually used to approximate our unknown function u. And that leads to the following equation we have now inside. We replace u bar by the term containing the sum over the basis functions. So, we have an integral over a sum over the basis functions and multiplying with now the test functions Phi j, and also on the right-hand side, we replace v by Phi j. Actually, to some extent, we're projecting the solution onto that set of basis functions and that will lead to conditions that will lead to a linear system of equations that we will discuss later. This is called the Galerkin principle, going back to a famous Russian engineer called Galerkin, you can see him here, lived until 1945. This is basically the principle that allows us to come up with a solution strategy for such a Finite-Element type problem.
