A numerical study of the spectral properties of Isogeometric collocation matrices for acoustic wave problems

Abstract: This paper focuses on the spectral properties of the mass and stiffness matrices for acoustic wave problems discretized with Isogeometric analysis (IGA) collocation methods in space and Newmark methods in time. Extensive numerical results are reported for the eigenvalues and condition numbers of the acoustic mass and stiffness matrices in the reference square domain with Dirichlet, Neumann and absorbing boundary conditions, varying the polynomial degree $p$, mesh size $h$, regularity $k$, of the IGA discretization and the time step $\Delta t$ and parameter $\beta$ of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the degrees of freedom d.o.f. and the number of nonzero entries nz are also reported. The results are comparable with the available spectral estimates for IGA Galerkin matrices associated to the Poisson problem with Dirichlet boundary conditions, and in some cases the IGA collocation results are better than the corresponding IGA Galerkin estimates.