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

TL;DR

This study investigates the spectral characteristics of matrices arising from Isogeometric collocation methods applied to acoustic wave problems, analyzing eigenvalues, condition numbers, and matrix sparsity through extensive numerical experiments.

Contribution

It provides a comprehensive numerical analysis of the spectral properties of IGA collocation matrices for acoustic problems, comparing results with existing spectral estimates and highlighting cases where collocation performs better.

Findings

Eigenvalues and condition numbers vary with polynomial degree, mesh size, and boundary conditions.

IGA collocation matrices often outperform Galerkin estimates in spectral properties.

Sparsity and eigenvalue distribution depend on degrees of freedom and matrix nonzero entries.

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.