My topic is going to be the high performance implementation of high-order finite element

discretization.

So I'm focusing on a particular topic of PDEs.

And I would like to start with a few applications that I've been active in and try to convince

you that the approach we're choosing is interesting and helpful.

So let me just give you three pictures here that's from simulations that we've done in

various areas.

On the left-hand side, you see a typical turbulent flow.

And what you see here, what probably is just noise on your screen on the left, is the very

fine scale structure that we are having in this flow.

And this means, of course, that we need high performance capabilities.

This is a simulation which is run on a supercomputer.

In the middle, you see a part of the human lung.

So what you see in the upper part of the figure is that this is some airways where we have

air going through.

And then it splits up into small structures.

And again, this is a typical problem that we want to solve in the order of understanding

of what happens in various scenarios.

And then on the right-hand side, I have a case of acoustics, which is some sound wave

which propagate in kind of a building and where we have been considering various optimizations

as well.

And all of these applications have in common that we are interested in solving them efficiently

and using the available hardware to a good extent.

So I want just to have two slides here on the general outline and ideas here.

So what I'm focusing on is the efficiency of the discretization.

Of course, what we want to do is we want to lower the number of degrees of freedom because

we are solving kind of the problems which are on the edge of what you could do even

on supercomputers.

So what we are aiming for is we want to have a higher accuracy per unknown that we have

in our method and the second ingredient that we try to get into is that we have a method

which should be applicable to general geometries.

So we are solving possibly complicated features that we need to resolve with unstructured

mesh.

And now of course the challenge, what I already said is that we have high Reynolds number

flows which develop fine-scale features.

So we need to track them over long times.

We need good dispersion properties because we want our methods to be resolving them well.

And the solution to these demands can be seen as a high-order finite element approximation

is a suitable case because they are geometric flexible because we can use unstructured meshes

and get high orders.

We prefer hexahedral meshes because we have some optimizations for them.

And the good thing about these kind of finite element type methods is that when we go to

higher orders we are adding unknowns inside of the elements.

So of course this means that we are giving up some of the flexibility in the mesh because

we are not having as many elements for the same number of unknowns but we still have

the ability to do unstructured meshes together.

And the degrees that we typically look at is between three and seven I say.

It might be depending on various applications where we do.

And I did say before we are looking at finite element type methods.

We often look at discontinuous Galerkin methods because they have attractive properties in