So welcome to the final weeks of the course where we now look at a method called the spectral element method that really takes all the best things from all of the methods we've investigated so far and puts them together to a very, very powerful numerical scheme. Actually, in the background, you see an example of a simulation using the spectral element method. This was developed and made by my colleagues at Princeton University in the U.S.. It shows the seismic wave field actually of a very damaging earthquake in Tohoku in 2011 and you see wave fields propagating away from the source region near Japan all over the globe. Now, as an example, this is developed on a very heterogeneous grid. It uses curved elements. These elements are actually adapted to the spherical surface. So that's one of the wonderful properties of the finite element method. It basically shows some of this geometrical flexibility that finite elements have that we also make use of in spectral elements. Now, I said it's a clever combination of different methods. Well, first of all, as usual, for the time extrapolation, we use the finite difference type approximation. Then we make use of the Galerkin principle that we became familiar with during the week where we discussed the linear finite element method. And then, inside the element, we were making use of concepts that we looked at in the week when we discussed the pseudospectral method. Remember, we had exact interpolation on the entire domain. Now we're using exact interpolation with spectral convergence inside an element. For example, making use of basis functions like the Lagrange basis functions. There are also other options. Like Chebyshev polynomials can also be used. But something miraculous happens, by combining it with an appropriate integration scheme, we will actually be able to have a diagonal mass matrix. Now, remember the mass matrix has to be inverted. If that's like a full matrix, it's extremely difficult to invert such a very, very big matrix as it is for realistic situations. If that matrix, then, is diagonal, actually, it's trivial to invert it. And that makes it very, very easy to parallelize. That's one of the reasons why, in my field in geophysics and in particular in seismology for elastic wave propagation, that method actually has been a very successful story ever since it was introduced maybe 15-20 years ago. So let's go into the maths. Of course, basically, the spectral element method is a flavor of the finite element method with specific basis functions inside the element. But to make this week and also the topic, the spectral element method, self-contained, we'll start again from scratch, develop the weak form, and then we need the concept of Lagrange interpolation for our displacement field inside the elements. And also we need the concept of numerical integration that we did not use in the finite element chapter because, remember the integrals, we actually assumed that the properties like density or shear modulus were constant inside an element. We want to relax this for the spectral element method. And actually, that is also one of the reasons why it will be so powerful. But for that, we have to actually discuss the problem of numerical integration. So let's start again from scratch and develope the whole maths and then put everything together to a powerful numerical scheme solving our differential equation.