Abstract
The conditioning and spectral properties of the mass and stiffness matrices for acoustic wave problems are here investigated when isogeometric analysis (IGA) collocation methods in space and Newmark methods in time are employed. Theoretical estimates and 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. This study focuses in particular on the spectral dependence on the polynomial degree p, mesh size h, regularity k, of the IGA discretization and on the time step size \(\Delta t\) and parameter \(\beta \) of the Newmark method. Results on the sparsity of the matrices and the eigenvalue distribution with respect to the number of degrees of freedom d.o.f. and the number of nonzero entries nz are also reported. The results show that the spectral properties of the IGA collocation matrices are comparable with the available spectral estimates for IGA Galerkin matrices associated with the Poisson problem with Dirichlet boundary conditions, and in some cases, the IGA collocation results are better than the corresponding IGA Galerkin estimates, in particular for increasing p and maximal regularity \(k=p-1\).
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Funding
Open access funding provided by Università degli Studi di Milano within the CRUI-CARE Agreement. This work was partially supported by the European Research Council through the FP7 Ideas Consolidator Grant HIGEOM n. 616563, by the Italian Ministry of Education, University and Research (MIUR) through the “Dipartimenti di Eccellenza Program 2018-22 - Dept. of Mathematics, University of Pavia,” and by the Istituto Nazionale di Alta Matematica (INdAM - GNCS), Italy.
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Communicated by: Lourenco Beirao da Veiga
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Zampieri, E., Pavarino, L.F. Conditioning and spectral properties of isogeometric collocation matrices for acoustic wave problems. Adv Comput Math 50, 16 (2024). https://doi.org/10.1007/s10444-024-10113-y
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DOI: https://doi.org/10.1007/s10444-024-10113-y
Keywords
- Acoustic waves
- Absorbing boundary conditions
- Isogeometric analysis
- Collocation
- Newmark method
- Condition number
- Spectral properties