Since my Phd, I am working on the Isogeometric Analysis (IGA) approach to solve (system of) partial differential equations using Finite Elements method.

IGA was introduced by T.J.R Hughes in 2005 with the aim of linking CAD and FEA. In the traditional workflow of Finite Elements Analysis (FEA), models are created from CAD representations. However, fixing CAD geometry and creating FEA models may account for more than 80% of overall analysis time and is a major engineering bottleneck.

The following picture gives us an idea on how IGA extends the traditional FEA approach, by using basis functions that are used in CAD systems.

There are many advantages of using B-Splines (and other CAD tools):

- High regularity: it has been shown, through applications but also theoretically, that high regular basis functions lead to interesting spectral properties. In fact, the number of ‘spurious’ branches in the spectrum depends on the regularity, and is reduced to 0 when the basis functions are of maximum regularity, which is the case of B-Splines.
- B-Splines have a very interesting algebraic property allowing for the construction of exact DeRham sequences and more generally it fits perfectly in the framework of the Finite Elements Exterior Calculus as introduced by Arnold and his coauthors [pdf].
- Better CFL numbers when having implicit time schemes
- Reduce the noise of Particle In Cell methods

As an example of the use of FEEC, we consider a hybrid model (1d-3v) for which we compare a standard Finite Elements method to the FEEC approach. In the following plot, we show simulated dispersion relation (linear regime) of the cold plasma (left) standard FEM (right) FEEC. As we can see, the geometric scheme behaves better for large $k$.

The next plot shows the energy conservation of the cold plasma coupled with a Vlasov equation (1d in space and 3d in velocity) in the non-linear regime

Right now, we are working on the spectra analysis of the discrete operators in the FEEC approach. We are also willing to extend our model to a Drift-kinetic-MHD hybrid model in general geometries.