{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T03:10:02Z","timestamp":1772248202160,"version":"3.50.1"},"reference-count":39,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2025,6,24]],"date-time":"2025-06-24T00:00:00Z","timestamp":1750723200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2025,6,24]],"date-time":"2025-06-24T00:00:00Z","timestamp":1750723200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100007642","name":"Universit\u00e0 degli Studi di Roma Tor Vergata","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100007642","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Numer. Math."],"published-print":{"date-parts":[[2025,8]]},"abstract":"<jats:title>Abstract<\/jats:title>\n          <jats:p>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 <jats:italic>p<\/jats:italic> 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 <jats:italic>d<\/jats:italic>-dimensional box. As a side result, we have an algebraic confirmation that the considered optimal and reduced spline spaces are indeed outlier-free.<\/jats:p>","DOI":"10.1007\/s00211-025-01470-x","type":"journal-article","created":{"date-parts":[[2025,6,23]],"date-time":"2025-06-23T23:03:19Z","timestamp":1750719799000},"page":"1397-1448","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Outlier-free isogeometric discretizations for Laplace eigenvalue problems: closed-form eigenvalue and eigenvector expressions"],"prefix":"10.1007","volume":"157","author":[{"given":"Noureddine","family":"Lamsahel","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carla","family":"Manni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Ahmed","family":"Ratnani","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefano","family":"Serra-Capizzano","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hendrik","family":"Speleers","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,6,24]]},"reference":[{"key":"1470_CR1","doi-asserted-by":"publisher","first-page":"125","DOI":"10.1080\/03081089108818063","volume":"29","author":"RB Bapat","year":"1991","unstructured":"Bapat, R.B., Sunder, V.S.: On hypergroups of matrices. Linear and Multilinear Algebra 29, 125\u2013140 (1991)","journal-title":"Linear and Multilinear Algebra"},{"key":"1470_CR2","doi-asserted-by":"publisher","first-page":"113","DOI":"10.1553\/etna_vol53s113","volume":"53","author":"G Barbarino","year":"2020","unstructured":"Barbarino, G., Garoni, C., Serra-Capizzano, S.: Block generalized locally Toeplitz sequences: Theory and applications in the multidimensional case. Electronic Transactions on Numerical Analysis 53, 113\u2013216 (2020)","journal-title":"Electronic Transactions on Numerical Analysis"},{"key":"1470_CR3","doi-asserted-by":"publisher","first-page":"99","DOI":"10.1016\/0024-3795(83)90009-5","volume":"52","author":"D Bini","year":"1983","unstructured":"Bini, D., Capovani, M.: Spectral and computational properties of band symmetric Toeplitz matrices. Linear Algebra and its Applications 52, 99\u2013126 (1983)","journal-title":"Linear Algebra and its Applications"},{"key":"1470_CR4","doi-asserted-by":"publisher","first-page":"136","DOI":"10.1553\/etna_vol58s136","volume":"58","author":"M Bolten","year":"2023","unstructured":"Bolten, M., Ekstr\u00f6m, S.-E., Furci, I., Serra-Capizzano, S.: A note on the spectral analysis of matrix sequences via GLT momentary symbols: From all-at-once solution of parabolic problems to distributed fractional order matrices. Electronic Transactions on Numerical Analysis 58, 136\u2013163 (2023)","journal-title":"Electronic Transactions on Numerical Analysis"},{"key":"1470_CR5","doi-asserted-by":"publisher","first-page":"312","DOI":"10.1137\/S0895479893245103","volume":"16","author":"E Bozzo","year":"1995","unstructured":"Bozzo, E., Di Fiore, C.: On the use of certain matrix algebras associated with discrete trigonometric transforms in matrix displacement decomposition. SIAM Journal on Matrix Analysis and Applications 16, 312\u2013326 (1995)","journal-title":"SIAM Journal on Matrix Analysis and Applications"},{"key":"1470_CR6","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1007\/s00211-019-01063-5","volume":"143","author":"A Bressan","year":"2019","unstructured":"Bressan, A., Sande, E.: Approximation in FEM, DG and IGA: A theoretical comparison. Numerische Mathematik 143, 923\u2013942 (2019)","journal-title":"Numerische Mathematik"},{"key":"1470_CR7","doi-asserted-by":"publisher","first-page":"275","DOI":"10.1016\/0024-3795(76)90101-4","volume":"13","author":"P Butler","year":"1976","unstructured":"Butler, P., Cantoni, A.: Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear Algebra and its Applications 13, 275\u2013288 (1976)","journal-title":"Linear Algebra and its Applications"},{"key":"1470_CR8","doi-asserted-by":"publisher","DOI":"10.1002\/9780470749081","volume-title":"Isogeometric Analysis: Toward Integration of CAD and FEA","author":"JA Cottrell","year":"2009","unstructured":"Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. John Wiley & Sons, Chichester (2009)"},{"key":"1470_CR9","doi-asserted-by":"publisher","first-page":"5257","DOI":"10.1016\/j.cma.2005.09.027","volume":"195","author":"JA Cottrell","year":"2006","unstructured":"Cottrell, J.A., Reali, A., Bazilevs, Y., Hughes, T.J.R.: Isogeometric analysis of structural vibrations. Computer Methods in Applied Mechanics and Engineering 195, 5257\u20135296 (2006)","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"1470_CR10","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4612-6333-3","volume-title":"A Practical Guide to Splines","author":"C de Boor","year":"1978","unstructured":"de Boor, C.: A Practical Guide to Splines, vol. 27. Springer-Verlag, New York (1978)"},{"key":"1470_CR11","doi-asserted-by":"publisher","first-page":"240","DOI":"10.1515\/spma-2020-0135","volume":"9","author":"Q Deng","year":"2021","unstructured":"Deng, Q.: Analytical solutions to some generalized and polynomial eigenvalue problems. Special Matrices 9, 240\u2013256 (2021)","journal-title":"Special Matrices"},{"key":"1470_CR12","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.113907","volume":"383","author":"Q Deng","year":"2021","unstructured":"Deng, Q., Calo, V.M.: A boundary penalization technique to remove outliers from isogeometric analysis on tensor-product meshes. Computer Methods in Applied Mechanics and Engineering 383, 113907 (2021)","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"1470_CR13","doi-asserted-by":"publisher","first-page":"35","DOI":"10.1080\/03081080008818658","volume":"48","author":"F Di Benedetto","year":"2000","unstructured":"Di Benedetto, F., Serra-Capizzano, S.: Optimal multilevel matrix algebra operators. Linear and Multilinear Algebra 48, 35\u201366 (2000)","journal-title":"Linear and Multilinear Algebra"},{"key":"1470_CR14","doi-asserted-by":"publisher","first-page":"49","DOI":"10.1016\/0024-3795(93)00347-3","volume":"229","author":"C Di Fiore","year":"1995","unstructured":"Di Fiore, C., Zellini, P.: Matrix decompositions using displacement rank and classes of commutative matrix algebras. Linear Algebra and its Applications 229, 49\u201399 (1995)","journal-title":"Linear Algebra and its Applications"},{"key":"1470_CR15","volume-title":"Kolmogorov $$n$$-width e spazi spline ottimi","author":"E Di Vona","year":"2019","unstructured":"Di Vona, E.: Kolmogorov $$n$$-width e spazi spline ottimi. Universit\u00e0 degli Studi di Roma Tor Vergata, Tesi di Laura Magistrale (2019)"},{"key":"1470_CR16","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1090\/mcom\/3027","volume":"85","author":"M Donatelli","year":"2016","unstructured":"Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Spectral analysis and spectral symbol of matrices in isogeometric collocation methods. Mathematics of Computation 85, 1639\u20131680 (2016)","journal-title":"Mathematics of Computation"},{"key":"1470_CR17","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1137\/140988590","volume":"55","author":"M Donatelli","year":"2017","unstructured":"Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis. SIAM Journal on Numerical Analysis 55, 31\u201362 (2017)","journal-title":"SIAM Journal on Numerical Analysis"},{"key":"1470_CR18","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2198","volume":"25","author":"S-E Ekstr\u00f6m","year":"2018","unstructured":"Ekstr\u00f6m, S.-E., Furci, I., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Are the eigenvalues of the B-spline isogeometric analysis approximation of $$-\\Delta u= \\lambda u$$ known in almost closed form? Numerical Linear Algebra with Applications 25, e2198 (2018)","journal-title":"Numerical Linear Algebra with Applications"},{"key":"1470_CR19","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.jat.2016.12.002","volume":"216","author":"MS Floater","year":"2017","unstructured":"Floater, M.S., Sande, E.: Optimal spline spaces of higher degree for $$L^2$$$$n$$-widths. Journal of Approximation Theory 216, 1\u201315 (2017)","journal-title":"Journal of Approximation Theory"},{"key":"1470_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00365-018-9427-5","volume":"50","author":"MS Floater","year":"2019","unstructured":"Floater, M.S., Sande, E.: Optimal spline spaces for $$L^2$$$$n$$-width problems with boundary conditions. Constructive Approximation 50, 1\u201318 (2019)","journal-title":"Constructive Approximation"},{"key":"1470_CR21","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1007\/s00211-013-0600-2","volume":"127","author":"C Garoni","year":"2014","unstructured":"Garoni, C., Manni, C., Pelosi, F., Serra-Capizzano, S., Speleers, H.: On the spectrum of stiffness matrices arising from isogeometric analysis. Numerische Mathematik 127, 751\u2013799 (2014)","journal-title":"Numerische Mathematik"},{"key":"1470_CR22","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115551","volume":"400","author":"C Garoni","year":"2022","unstructured":"Garoni, C., Manni, C., Pelosi, F., Speleers, H.: Spectral analysis of matrices resulting from isogeometric immersed methods and trimmed geometries. Computer Methods in Applied Mechanics and Engineering 400, 115551 (2022)","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"1470_CR23","doi-asserted-by":"publisher","DOI":"10.1002\/nla.2318","volume":"27","author":"C Garoni","year":"2020","unstructured":"Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: NURBS in isogeometric discretization methods: A spectral analysis. Numerical Linear Algebra with Applications 27, e2318 (2020)","journal-title":"Numerical Linear Algebra with Applications"},{"key":"1470_CR24","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-53679-8","volume-title":"Generalized Locally Toeplitz Sequences: Theory and Applications","author":"C Garoni","year":"2017","unstructured":"Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol. 1. Springer, Cham (2017)"},{"key":"1470_CR25","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-02233-4","volume-title":"Generalized Locally Toeplitz Sequences: Theory and Applications","author":"C Garoni","year":"2018","unstructured":"Garoni, C., Serra-Capizzano, S.: Generalized Locally Toeplitz Sequences: Theory and Applications, vol. 2. Springer, Cham (2018)"},{"key":"1470_CR26","doi-asserted-by":"publisher","first-page":"1100","DOI":"10.1137\/140976480","volume":"36","author":"C Garoni","year":"2015","unstructured":"Garoni, C., Serra-Capizzano, S., Sesana, D.: Spectral analysis and spectral symbol of $$d$$-variate $$Q_p$$ Lagrangian FEM stiffness matrices. SIAM Journal on Matrix Analysis and Applications 36, 1100\u20131128 (2015)","journal-title":"SIAM Journal on Matrix Analysis and Applications"},{"key":"1470_CR27","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1007\/s11831-018-9295-y","volume":"26","author":"C Garoni","year":"2019","unstructured":"Garoni, C., Speleers, H., Ekstr\u00f6m, S.-E., Reali, A., Serra-Capizzano, S., Hughes, T.J.R.: Symbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: Exposition and review. Archives of Computational Methods in Engineering 26, 1639\u20131690 (2019)","journal-title":"Archives of Computational Methods in Engineering"},{"key":"1470_CR28","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114115","volume":"387","author":"RR Hiemstra","year":"2021","unstructured":"Hiemstra, R.R., Hughes, T.J.R., Reali, A., Schillinger, D.: Removal of spurious outlier frequencies and modes from isogeometric discretizations of second- and fourth-order problems in one, two, and three dimensions. Computer Methods in Applied Mechanics and Engineering 387, 114115 (2021)","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"1470_CR29","doi-asserted-by":"publisher","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","volume":"194","author":"TJR Hughes","year":"2005","unstructured":"Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Computer Methods in Applied Mechanics and Engineering 194, 4135\u20134195 (2005)","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"1470_CR30","doi-asserted-by":"publisher","first-page":"290","DOI":"10.1016\/j.cma.2013.11.012","volume":"272","author":"TJR Hughes","year":"2014","unstructured":"Hughes, T.J.R., Evans, J.A., Reali, A.: Finite element and NURBS approximations of eigenvalue, boundary-value, and initial-value problems. Computer Methods in Applied Mechanics and Engineering 272, 290\u2013320 (2014)","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"1470_CR31","series-title":"Lecture Notes in Mathematics","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/978-3-319-94911-6_1","volume-title":"Splines and PDEs: From Approximation Theory to Numerical Linear Algebra","author":"T Lyche","year":"2018","unstructured":"Lyche, T., Manni, C., Speleers, H.: Foundations of spline theory: B-splines, spline approximation, and hierarchical refinement. In: Lyche, T., Manni, C., Speleers, H. (eds.) Splines and PDEs: From Approximation Theory to Numerical Linear Algebra. Lecture Notes in Mathematics, vol. 2219, pp. 1\u201376. Springer, Cham (2018)"},{"key":"1470_CR32","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2021.114260","volume":"389","author":"C Manni","year":"2022","unstructured":"Manni, C., Sande, E., Speleers, H.: Application of optimal spline subspaces for the removal of spurious outliers in isogeometric discretizations. Computer Methods in Applied Mechanics and Engineering 389, 114260 (2022)","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"1470_CR33","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2023.116314","volume":"417","author":"C Manni","year":"2023","unstructured":"Manni, C., Sande, E., Speleers, H.: Outlier-free spline spaces for isogeometric discretizations of biharmonic and polyharmonic eigenvalue problems. Computer Methods in Applied Mechanics and Engineering 417, 116314 (2023)","journal-title":"Computer Methods in Applied Mechanics and Engineering"},{"key":"1470_CR34","doi-asserted-by":"publisher","first-page":"1175","DOI":"10.1142\/S0218202519500192","volume":"29","author":"E Sande","year":"2019","unstructured":"Sande, E., Manni, C., Speleers, H.: Sharp error estimates for spline approximation: Explicit constants, $$n$$-widths, and eigenfunction convergence. Mathematical Models and Methods in Applied Sciences 29, 1175\u20131205 (2019)","journal-title":"Mathematical Models and Methods in Applied Sciences"},{"key":"1470_CR35","doi-asserted-by":"publisher","first-page":"889","DOI":"10.1007\/s00211-019-01097-9","volume":"144","author":"E Sande","year":"2020","unstructured":"Sande, E., Manni, C., Speleers, H.: Explicit error estimates for spline approximation of arbitrary smoothness in isogeometric analysis. Numerische Mathematik 144, 889\u2013929 (2020)","journal-title":"Numerische Mathematik"},{"key":"1470_CR36","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511618994","volume-title":"Spline Functions: Basic Theory","author":"LL Schumaker","year":"2007","unstructured":"Schumaker, L.L.: Spline Functions: Basic Theory. Cambridge University Press, Cambridge (2007)"},{"key":"1470_CR37","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/S0898-1221(00)00072-9","volume":"39","author":"S Serra-Capizzano","year":"2000","unstructured":"Serra-Capizzano, S.: Some theorems on linear positive operators and functionals and their applications. Computers & Mathematics with Applications 39, 139\u2013167 (2000)","journal-title":"Computers & Mathematics with Applications"},{"key":"1470_CR38","doi-asserted-by":"publisher","first-page":"105","DOI":"10.1016\/j.camwa.2018.09.017","volume":"77","author":"J Sogn","year":"2019","unstructured":"Sogn, J., Takacs, S.: Robust multigrid solvers for the biharmonic problem in isogeometric analysis. Computers & Mathematics with Applications 77, 105\u2013124 (2019)","journal-title":"Computers & Mathematics with Applications"},{"key":"1470_CR39","doi-asserted-by":"publisher","first-page":"1411","DOI":"10.1142\/S0218202516500342","volume":"26","author":"S Takacs","year":"2016","unstructured":"Takacs, S., Takacs, T.: Approximation error estimates and inverse inequalities for B-splines of maximum smoothness. Mathematical Models and Methods in Applied Sciences 26, 1411\u20131445 (2016)","journal-title":"Mathematical Models and Methods in Applied Sciences"}],"container-title":["Numerische Mathematik"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01470-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00211-025-01470-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00211-025-01470-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,9,19]],"date-time":"2025-09-19T16:30:42Z","timestamp":1758299442000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00211-025-01470-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,6,24]]},"references-count":39,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2025,8]]}},"alternative-id":["1470"],"URL":"https:\/\/doi.org\/10.1007\/s00211-025-01470-x","relation":{},"ISSN":["0029-599X","0945-3245"],"issn-type":[{"value":"0029-599X","type":"print"},{"value":"0945-3245","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,6,24]]},"assertion":[{"value":"26 April 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 March 2025","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 April 2025","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"24 June 2025","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}