A comparative numerical study of spectral properties in isogeometric collocation and Galerkin methods for acoustic waves

We approximate the acoustic wave equation in two-dimensional regions using collocation and Galerkin isogeometric analysis (IGA) in space, coupled with implicit second-order Newmark schemes for time integration. We present a detailed numerical study that examines and compares the behavior of extreme eigenvalues and condition numbers of the mass and iteration IGA matrices, varying the polynomial degree p, mesh size h, regularity k, and the boundary conditions, that can be either Dirichlet or absorbing in order to simulate unbounded domains. We propose and validate numerically some conjectures related to the IGA collocation and Galerkin matrices for the wave equation with different types of boundary conditions, extending similar results that are known for the IGA Galerkin approximation, limitedly to the case of the Poisson problem with Dirichlet boundary conditions, and generalizing earlier results obtained within the framework of the collocation method. The results show that the spectral properties of the IGA collocation matrices are analogous and in most cases better than the corresponding IGA Galerkin discretization of the Poisson problem with Dirichlet or absorbing boundary conditions.