Study and use of spectral symbol properties for isogeometric matrices on trimmed geometries

We study the spectral behavior of (sequences of) matrices resulting from immersed isogeometric discretizations on trimmed geometries. They enjoy an asymptotic spectral distribution, described by a (spectral) symbol, and we discuss some properties of this symbol. In particular, we show that the structure and properties of the symbol are completely analogous to the untrimmed case when a suitable natural restriction of the parametric domain is considered. This spectral knowledge can be exploited to identify potentially fast preconditioners for the considered immersed discretization matrices and we propose a specific CG preconditioner based on the symbol. We also provide numerical experiments that support the correctness of the theoretical results and illustrate the performance of the proposed preconditioner.