Isogeometric methods in computational electromagnetism

A sequence of spline spaces satisfying a de Rham complex was first introduced in 2010 for their application in isogeometric analysis. These spaces can be seen as a generalization of hexahedral edge and face finite elements with higher inter-element continuity. They have been succesfully used in electromagnetics, fluid mechanics and magnetohydrodynamics.
In this talk I will start presenting in detail the construction of the spline de Rham complex in the tensor-product case, and the commutative projectors necessary for their analysis. Then, I will show how some of the properties from low order finite elements extend to isogeometric methods, thanks to commutative isomorphisms with finite elements defined in an auxiliary grid. Finally, I will briefly present results on recent topics and open problems regarding the extension to adaptive methods with hierarchical splines.