# Fast algorithms for tensor product discretizations in Isogeometric Analysis and beyond

## Dr. Clemens Hofreither

**June 26, 2020, 1 p.m. ZOOM**

In this talk, we consider discretizations of partial differential equations which have an underlying tensor

product structure (e.g., are posed in a square or cube domain) and ask the question: how can we

exploit this underlying structure in order to obtain faster algorithms for the generation and solution of

the discretized problem?

Although this assumption of tensor product structure is quite strong, there has recently been renewed

interest in such algorithms due to the rise of Isogeometric Analysis. This young competitor to the Finite

Element Method relies strongly on tensor product spline spaces and allows the treatment of more

complicated computational domains while preserving the underlying tensor product structure of the

discretization space.

We will consider various problems, such as the fast formation of the isogeometric stiffness matrices,

the development of multigrid solvers which are robust with respect to the spline degree, and the

construction of solvers which mitigate the exponential dependence on the space dimension, where the

tensor product structure can be exploited in order to achieve significant speedups over classical

techniques. In several situations, methods of low-rank matrix and tensor approximation come into

play.

Although the focus of the talk is on applications in Isogeometric Analysis, most of the developed

techniques are rather general and can be used for different discretization techniques. As a particularly

timely application, we will see some results for the solution of fractional diffusion problems in general

domains.