{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T04:32:46Z","timestamp":1770352366042,"version":"3.49.0"},"reference-count":34,"publisher":"American Mathematical Society (AMS)","issue":"358","license":[{"start":{"date-parts":[[2026,2,12]],"date-time":"2026-02-12T00:00:00Z","timestamp":1770854400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"funder":[{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["0.55776\/P33477"],"award-info":[{"award-number":["0.55776\/P33477"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["https:\/\/doi.org\/10.55776\/F65"],"award-info":[{"award-number":["https:\/\/doi.org\/10.55776\/F65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["https:\/\/doi.org\/10.55776\/P33477"],"award-info":[{"award-number":["https:\/\/doi.org\/10.55776\/P33477"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["0.55776\/P33477"],"award-info":[{"award-number":["0.55776\/P33477"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["https:\/\/doi.org\/10.55776\/F65"],"award-info":[{"award-number":["https:\/\/doi.org\/10.55776\/F65"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100002428","name":"Austrian Science Fund","doi-asserted-by":"publisher","award":["https:\/\/doi.org\/10.55776\/P33477"],"award-info":[{"award-number":["https:\/\/doi.org\/10.55776\/P33477"]}],"id":[{"id":"10.13039\/501100002428","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"<p>We consider a family of conforming space\u2013time finite element discretizations for the wave equation based on splines of maximal regularity in time. Traditional techniques may require a Courant\u2013Friedrichs\u2013Lewy (CFL) condition to guarantee stability. Recent works by O. Steinbach and M. Zank (2019), and S. Fraschini, G. Loli, A. Moiola, and G. Sangalli (2024), have introduced unconditionally stable schemes by adding nonconsistent penalty terms to the underlying bilinear form. Stability and error analysis have been carried out for lowest order discrete spaces. While higher order methods have shown promising properties through numerical testing, their rigorous analysis was still missing. In this paper, we address this stability analysis by studying the properties of the condition number of a family of matrices associated with the time discretization. For each spline order, we derive explicit estimates of both the CFL condition required in the unstabilized case and the penalty term that minimises the consistency error in the stabilized case. Numerical tests confirm the sharpness of our results.<\/p>","DOI":"10.1090\/mcom\/4062","type":"journal-article","created":{"date-parts":[[2024,12,18]],"date-time":"2024-12-18T10:19:35Z","timestamp":1734517175000},"page":"683-719","source":"Crossref","is-referenced-by-count":4,"title":["Stability of conforming space\u2013time isogeometric methods for the wave equation"],"prefix":"10.1090","volume":"95","author":[{"given":"Matteo","family":"Ferrari","sequence":"first","affiliation":[]},{"given":"Sara","family":"Fraschini","sequence":"additional","affiliation":[]}],"member":"14","published-online":{"date-parts":[[2025,2,12]]},"reference":[{"issue":"10","key":"1","doi-asserted-by":"publisher","first-page":"29","DOI":"10.1016\/0895-7177(96)00052-0","article-title":"The conditioning of Toeplitz band matrices","volume":"23","author":"Amodio, P.","year":"1996","journal-title":"Math. Comput. Modelling","ISSN":"https:\/\/id.crossref.org\/issn\/0895-7177","issn-type":"print"},{"key":"2","unstructured":"P. Bignardi and A. Moiola, A space-time continuous and coercive formulation for the wave equation,  arXiv:2312.07268, 2023."},{"issue":"3-4","key":"3","doi-asserted-by":"publisher","first-page":"235","DOI":"10.1006\/jsco.1996.0125","article-title":"The Magma algebra system. I. The user language","volume":"24","author":"Bosma, Wieb","year":"1997","journal-title":"J. Symbolic Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0747-7171","issn-type":"print"},{"issue":"4","key":"4","doi-asserted-by":"publisher","first-page":"923","DOI":"10.1007\/s00211-019-01063-5","article-title":"Approximation in FEM, DG and IGA: a theoretical comparison","volume":"143","author":"Bressan, Andrea","year":"2019","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"5","series-title":"Wavelet Analysis and its Applications","isbn-type":"print","volume-title":"An introduction to wavelets","volume":"1","author":"Chui, Charles K.","year":"1992","ISBN":"https:\/\/id.crossref.org\/isbn\/0121745848"},{"issue":"1","key":"6","doi-asserted-by":"publisher","first-page":"31","DOI":"10.1137\/140988590","article-title":"Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis","volume":"55","author":"Donatelli, Marco","year":"2017","journal-title":"SIAM J. Numer. Anal.","ISSN":"https:\/\/id.crossref.org\/issn\/0036-1429","issn-type":"print"},{"issue":"5","key":"7","doi-asserted-by":"publisher","first-page":"e2198, 34","DOI":"10.1002\/nla.2198","article-title":"Are the eigenvalues of the B-spline isogeometric analysis approximation of -\u0394\ud835\udc62=\ud835\udf06\ud835\udc62 known in almost closed form?","volume":"25","author":"Ekstr\u00f6m, Sven-Erik","year":"2018","journal-title":"Numer. Linear Algebra Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/1070-5325","issn-type":"print"},{"issue":"21-26","key":"8","doi-asserted-by":"publisher","first-page":"1726","DOI":"10.1016\/j.cma.2009.01.021","article-title":"\ud835\udc5b-widths, sup-infs, and optimality ratios for the \ud835\udc58-version of the isogeometric finite element method","volume":"198","author":"Evans, John A.","year":"2009","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"9","unstructured":"S. Fraschini, Stability of space\u2013time isogeometric methods for wave propagation problems,  arXiv:2303.15460, 2023."},{"key":"10","doi-asserted-by":"publisher","first-page":"205","DOI":"10.1016\/j.camwa.2024.06.009","article-title":"An unconditionally stable space-time isogeometric method for the acoustic wave equation","volume":"169","author":"Fraschini, S.","year":"2024","journal-title":"Comput. Math. Appl.","ISSN":"https:\/\/id.crossref.org\/issn\/0898-1221","issn-type":"print"},{"key":"11","unstructured":"S. Fraschini, A. Moiola, and G. Sangalli, Stability of space\u2013time isogeometric methods for wave propagation problems, WAVES 2022 - The 15th International Conference on Mathematical and Numerical Aspects of Wave Propagation, 2022, pp. 286\u2013287."},{"issue":"4","key":"12","doi-asserted-by":"publisher","first-page":"751","DOI":"10.1007\/s00211-013-0600-2","article-title":"On the spectrum of stiffness matrices arising from isogeometric analysis","volume":"127","author":"Garoni, Carlo","year":"2014","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"issue":"5","key":"13","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1007\/s11831-018-9295-y","article-title":"Symbol-based analysis of finite element and isogeometric B-spline discretizations of eigenvalue problems: exposition and review","volume":"26","author":"Garoni, Carlo","year":"2019","journal-title":"Arch. Comput. Methods Eng.","ISSN":"https:\/\/id.crossref.org\/issn\/1134-3060","issn-type":"print"},{"key":"14","series-title":"Johns Hopkins Studies in the Mathematical Sciences","isbn-type":"print","volume-title":"Matrix computations","author":"Golub, Gene H.","year":"1996","ISBN":"https:\/\/id.crossref.org\/isbn\/080185413X","edition":"3"},{"key":"15","isbn-type":"print","volume-title":"Table of integrals, series, and products","author":"Gradshteyn, I. S.","year":"2015","ISBN":"https:\/\/id.crossref.org\/isbn\/9780123849335","edition":"8"},{"issue":"39-41","key":"16","doi-asserted-by":"publisher","first-page":"4135","DOI":"10.1016\/j.cma.2004.10.008","article-title":"Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement","volume":"194","author":"Hughes, T. J. R.","year":"2005","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"3","key":"17","doi-asserted-by":"publisher","first-page":"339","DOI":"10.1016\/0045-7825(88)90006-0","article-title":"Space-time finite element methods for elastodynamics: formulations and error estimates","volume":"66","author":"Hughes, Thomas J. R.","year":"1988","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"49-50","key":"18","doi-asserted-by":"publisher","first-page":"4104","DOI":"10.1016\/j.cma.2008.04.006","article-title":"Duality and unified analysis of discrete approximations in structural dynamics and wave propagation: comparison of \ud835\udc5d-method finite elements with \ud835\udc58-method NURBS","volume":"197","author":"Hughes, T. J. R.","year":"2008","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"341","key":"19","doi-asserted-by":"publisher","first-page":"1211","DOI":"10.1090\/mcom\/3786","article-title":"A space-time quasi-Trefftz DG method for the wave equation with piecewise-smooth coefficients","volume":"92","author":"Imbert-G\u00e9rard, Lise-Marie","year":"2023","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"issue":"3-4","key":"20","doi-asserted-by":"publisher","first-page":"645","DOI":"10.5486\/pmd.2002.2756","article-title":"On zeros of reciprocal polynomials","volume":"61","author":"Lakatos, Piroska","year":"2002","journal-title":"Publ. Math. Debrecen","ISSN":"https:\/\/id.crossref.org\/issn\/0033-3883","issn-type":"print"},{"key":"21","series-title":"Mathematics in Science and Engineering","isbn-type":"print","volume-title":"Theory of difference equations","volume":"181","author":"Lakshmikantham, V.","year":"1988","ISBN":"https:\/\/id.crossref.org\/isbn\/0124341004"},{"key":"22","isbn-type":"print","doi-asserted-by":"publisher","first-page":"625","DOI":"10.1007\/978-3-030-95025-5\\_68","article-title":"Numerical results for an unconditionally stable space-time finite element method for the wave equation","author":"L\u00f6scher, Richard","year":"[2022] \\copyright2022","ISBN":"https:\/\/id.crossref.org\/isbn\/9783030950248"},{"key":"23","isbn-type":"print","volume-title":"Chebyshev polynomials","author":"Mason, J. C.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0849303559"},{"issue":"2","key":"24","doi-asserted-by":"publisher","first-page":"389","DOI":"10.1007\/s00211-017-0910-x","article-title":"A space-time Trefftz discontinuous Galerkin method for the acoustic wave equation in first-order formulation","volume":"138","author":"Moiola, Andrea","year":"2018","journal-title":"Numer. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0029-599X","issn-type":"print"},{"key":"25","doi-asserted-by":"publisher","first-page":"443","DOI":"10.1007\/s10915-004-4132-5","article-title":"A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media","volume":"22\/23","author":"Monk, Peter","year":"2005","journal-title":"J. Sci. Comput.","ISSN":"https:\/\/id.crossref.org\/issn\/0885-7474","issn-type":"print"},{"key":"26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1016\/j.cam.2018.10.049","article-title":"A double inequality for the ratio of two non-zero neighbouring Bernoulli numbers","volume":"351","author":"Qi, Feng","year":"2019","journal-title":"J. Comput. Appl. Math.","ISSN":"https:\/\/id.crossref.org\/issn\/0377-0427","issn-type":"print"},{"key":"27","series-title":"Pure and Applied Mathematics (New York)","isbn-type":"print","volume-title":"Chebyshev polynomials","author":"Rivlin, Theodore J.","year":"1990","ISBN":"https:\/\/id.crossref.org\/isbn\/0471628964","edition":"2"},{"key":"28","series-title":"Oxford Applied Mathematics and Computing Science Series","isbn-type":"print","volume-title":"Numerical solution of partial differential equations","author":"Smith, G. D.","year":"1985","ISBN":"https:\/\/id.crossref.org\/isbn\/0198596413","edition":"3"},{"key":"29","isbn-type":"print","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1007\/978-3-030-14244-5\\_17","article-title":"A stabilized space-time finite element method for the wave equation","author":"Steinbach, Olaf","year":"[2019] \\copyright2019","ISBN":"https:\/\/id.crossref.org\/isbn\/9783030142445"},{"key":"30","doi-asserted-by":"publisher","first-page":"154","DOI":"10.1553\/etna\\_vol52s154","article-title":"Coercive space-time finite element methods for initial boundary value problems","volume":"52","author":"Steinbach, Olaf","year":"2020","journal-title":"Electron. Trans. Numer. Anal."},{"key":"31","unstructured":"The MathWorks Inc., Symbolic Math Toolbox, Natick, Massachusetts, United States, 2019."},{"key":"32","doi-asserted-by":"crossref","unstructured":"R. Vieira, Polynomials with symmetric zeros, Polynomials (Cheon Seoung Ryoo, ed.), IntechOpen, Rijeka, 2019.","DOI":"10.5772\/intechopen.82728"},{"key":"33","unstructured":"M. Zank, Inf-Sup Stable Space\u2013Time Methods for Time-Dependent Partial Differential Equations, Verlag d. Technischen Universit\u00e4t Graz, 2020."},{"key":"34","doi-asserted-by":"crossref","unstructured":"M. Zank, Higher-order space\u2013time continuous Galerkin methods for the wave equation, 14th WCCM-ECCOMAS Congress 2020, vol. 700, 2021.","DOI":"10.23967\/wccm-eccomas.2020.167"}],"container-title":["Mathematics of Computation"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-358\/S0025-5718-2025-04062-6\/mcom4062_AM.pdf","content-type":"application\/pdf","content-version":"am","intended-application":"syndication"},{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-358\/S0025-5718-2025-04062-6\/S0025-5718-2025-04062-6.pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,9]],"date-time":"2025-12-09T15:45:18Z","timestamp":1765295118000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.ams.org\/mcom\/2026-95-358\/S0025-5718-2025-04062-6\/"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,2,12]]},"references-count":34,"journal-issue":{"issue":"358","published-print":{"date-parts":[[2026,3]]}},"alternative-id":["S0025-5718-2025-04062-6"],"URL":"https:\/\/doi.org\/10.1090\/mcom\/4062","archive":["CLOCKSS","Portico"],"relation":{},"ISSN":["1088-6842","0025-5718"],"issn-type":[{"value":"1088-6842","type":"electronic"},{"value":"0025-5718","type":"print"}],"subject":[],"published":{"date-parts":[[2025,2,12]]}}}