Usually, the finite element method employs globally continuous, piecewise polynomials on regular triangulations to approximate the solution of a given partial differential equation (PDE). In particular, the problem geometry must be first approximated by a polygon/polyhedron and then triangulated.
In practice, such a domain stems essentially always from a CAD-program. The idea of the so-called isogeometric analysis is to use the same functions for the approximation of the solution as are used for the representation of the geometry. Therefore, the geometry does not have to be approximated nor meshed. Usually, so-called splines are used for the geometry representation in CAD. These are defined as tensor product of one-dimensional splines which are polynomials with certain differentiability properties at the grid points. However, this approach works only on tensor grids. To allow for adaptive refinement, e.g., to resolve possible singularities of the PDE solution, several extensions such as hierarchical splines, T-splines or LR-splines have recently been developed.
The lecture introduces the participants to the current state of research, so that they could participate in ongoing research activities.
