Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions
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arXiv
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| Format: | Preprint |
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2025
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| author | Lamsahel, Noureddine Manni, Carla Ratnani, Ahmed Serra-Capizzano, Stefano Speleers, Hendrik |
| author_facet | Lamsahel, Noureddine Manni, Carla Ratnani, Ahmed Serra-Capizzano, Stefano Speleers, Hendrik |
| contents | We derive explicit closed-form expressions for the eigenvalues and eigenvectors of the matrices resulting from isogeometric Galerkin discretizations based on outlier-free spline subspaces for the Laplace operator, under different types of homogeneous boundary conditions on bounded intervals. For optimal spline subspaces and specific reduced spline spaces, represented in terms of B-spline-like bases, we show that the corresponding mass and stiffness matrices exhibit a Toeplitz-minus-Hankel or Toeplitz-plus-Hankel structure. Such matrix structure holds for any degree p and implies that the eigenvalues are an explicitly known sampling of the spectral symbol of the Toeplitz part. Moreover, by employing tensor-product arguments, we extend the closed-form property of the eigenvalues and eigenvectors to a d-dimensional box. As a side result, we have an algebraic confirmation that the considered optimal and reduced spline spaces are indeed outlier-free. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_01487 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions Lamsahel, Noureddine Manni, Carla Ratnani, Ahmed Serra-Capizzano, Stefano Speleers, Hendrik Numerical Analysis Spectral Theory We derive explicit closed-form expressions for the eigenvalues and eigenvectors of the matrices resulting from isogeometric Galerkin discretizations based on outlier-free spline subspaces for the Laplace operator, under different types of homogeneous boundary conditions on bounded intervals. For optimal spline subspaces and specific reduced spline spaces, represented in terms of B-spline-like bases, we show that the corresponding mass and stiffness matrices exhibit a Toeplitz-minus-Hankel or Toeplitz-plus-Hankel structure. Such matrix structure holds for any degree p and implies that the eigenvalues are an explicitly known sampling of the spectral symbol of the Toeplitz part. Moreover, by employing tensor-product arguments, we extend the closed-form property of the eigenvalues and eigenvectors to a d-dimensional box. As a side result, we have an algebraic confirmation that the considered optimal and reduced spline spaces are indeed outlier-free. |
| title | Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions |
| topic | Numerical Analysis Spectral Theory |
| url | https://arxiv.org/abs/2505.01487 |