So, in our example we investigated the behavior of the finite element versus finite difference method in homogeneous medium. Now, one of the key advantages and the power of finite element method actually that it's very elegant and very easy to change the element size. We want to investigate that in a simple example. If you think of, again I'm a seismologist, if you think of an Earth model, of course the seismic velocities inside the Earth varies with space and there's an example behind me shown where actually the color scheme indicates that the velocity varies in space and we would like to adapt our mesh, our computational grid to the situation. To make it simple, let's look at this in a one-dimensional example. Here's a one-dimensional domain and at the center is a low velocity zone, let's call it like this, with 1,500 meters per second and to the left there is a domain with a velocity of 6,000 meters per seconds of factor of four different. On the right-hand side there is a velocity of 3,000 meters per second. So, what would happen if we discretize this entire domain with the same grid spacing, be it finite differences or finite elements? In finite elements, it will be the element size. Well, what happens is that we have to adapt our time step and we have to adapt our grid spacing according to the smallest velocity in which in this case would be 1,500 meters per second. So, what would happen, for example, if we assume we need 30 grid points or 30 elements per wavelength, we would then actually oversample fundamentally, dramatically, the wavefield in the space outside this domain. So, we would have a lot more grid points per wavelength. That would be a waste of memory, a waste of floating point operations, it would slow down the simulation and that's not what we want. That's why we now resort to something that's called h-adaptivity. The h of course stands for the size of the element and that's one of the key properties of potential of finite element type methods to adapt the h, adapt the elements size to the actual physical model. In this case, it corresponds to the seismic velocities but it might also be in fluid dynamics corresponding to flow velocities or other physical properties. So, let's find an optimal solution to this problem, adapting our element size. In this table, you see the properties again of the medium and also some of the simulation parameters for our finite element simulation. For example, the frequency range which then basically gives us the targeted wavelengths and by that we can adapt the element size h to the central low velocity zone, and in that case the element size actually is 10 meters. We adapt now the element size of the domains outside the low velocity zone to 40 meters or a factor of four for the medium with 6,000 meters per second, and 20 meters for the medium part with 3,000 meters per second. Now we can start the simulation. Actually you see that the wavefield is now developing. Actually the horizontal axis in this case is space and the vertical axis is time. You can see actually from the slope of the wavefronts emanating. The steeper the wavefronts are in that example, the slower are the velocities and the shallower the wavefronts actually the faster are the velocities. So, this is now extrapolated in time, time goes vertically up. You can see that the wavefield actually bounces back and forth inside that low velocity zone, and that's an example of basically a trapping of energy that we also see in nature and in many other physical phenomena where basically the reflection coefficients at those boundaries makes that the energy is preferably propagating inside this low velocity zone. So, that's an example that shows basically the power of the finite element scheme. It's very easy to adapt the mesh. It only basically changes the entries of the elements in the mass matrix and the stiffness matrix, all this is pre-calculated at the actual simulation time when he extrapolated the system, in time there is no difference to the algorithm at all. So, that's a very elegant way of using the so-called h-adaptivity. This plays a very, very big role when we have solutions for higher dimensions, two-dimensions and three-dimensions because otherwise actually you would not be able to solve realistic physical systems.