Spectral analysis of matrices resulting from isogeometric immersed methods and trimmed geometries

Dealing with complex and trimmed geometries is one of the current challenges in isogeometric analysis. Coupling the isogeometric paradigm with the framework of immersed boundary methods is an attractive practical strategy. In this paper, we consider general variable-coefficient Poisson problems and propose a new formulation of an immersed Galerkin discretization based on tensor-product cardinal B-splines. The weakly imposed boundary conditions do not require any tuning of user-defined penalty or stabilization parameters, and the system matrices are symmetric for symmetric problems. We analyze the spectral behavior of such matrices and prove that they enjoy an asymptotic spectral distribution when the matrix size tends to infinity. We provide an explicit description of this asymptotic distribution which turns out to have a canonical structure incorporating the coefficients of the differential operator and the discretization technique. In the spirit of the isogeometric paradigm, trimmed (single-patch) geometry maps can be treated by means of the same immersed approach. Therefore, we are able to describe spectral properties of matrices in the context of trimmed geometries as well. The theoretical findings are complemented with a selection of numerical experiments.