{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,30]],"date-time":"2026-04-30T00:22:50Z","timestamp":1777508570601,"version":"3.51.4"},"reference-count":30,"publisher":"American Mathematical Society (AMS)","issue":"305","license":[{"start":{"date-parts":[[2017,8,3]],"date-time":"2017-08-03T00:00:00Z","timestamp":1501718400000},"content-version":"am","delay-in-days":365,"URL":"https:\/\/www.ams.org\/publications\/copyright-and-permissions"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Math. Comp."],"abstract":"

\n A linear full elliptic second-order Partial Differential Equation (PDE), defined on a\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -dimensional domain\n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \\Omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , is approximated by the isogeometric Galerkin method based on uniform tensor-product B-splines of degrees\n \n \n \n \n (<\/mml:mo>\n \n p<\/mml:mi>\n 1<\/mml:mn>\n <\/mml:msub>\n ,<\/mml:mo>\n \n \u2026\n \n <\/mml:mo>\n ,<\/mml:mo>\n \n p<\/mml:mi>\n d<\/mml:mi>\n <\/mml:msub>\n )<\/mml:mo>\n <\/mml:mrow>\n (p_1,\\ldots ,p_d)<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n . The considered approximation process leads to a\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -level stiffness matrix, banded in a multilevel sense. This matrix is close to a\n \n \n \n d<\/mml:mi>\n d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n -level Toeplitz structure if the PDE coefficients are constant and the physical domain\n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \\Omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n is the hypercube\n \n \n \n \n (<\/mml:mo>\n 0<\/mml:mn>\n ,<\/mml:mo>\n 1<\/mml:mn>\n \n )<\/mml:mo>\n d<\/mml:mi>\n <\/mml:msup>\n <\/mml:mrow>\n (0,1)^d<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n without using any geometry map. In such a simplified case, a detailed spectral analysis of the stiffness matrices has already been carried out in a previous work. In this paper, we complete the picture by considering non-constant PDE coefficients and an arbitrary domain\n \n \n \n \n \u03a9\n \n <\/mml:mi>\n \\Omega<\/mml:annotation>\n <\/mml:semantics>\n <\/mml:math>\n <\/inline-formula>\n , parameterized with a non-trivial geometry map. We compute and study the spectral symbol of the related stiffness matrices. This symbol describes the asymptotic eigenvalue distribution when the fineness parameters tend to zero (so that the matrix-size tends to infinity). The mathematical tool used for computing the symbol is the theory of Generalized Locally Toeplitz (GLT) sequences.\n <\/p>","DOI":"10.1090\/mcom\/3143","type":"journal-article","created":{"date-parts":[[2016,2,9]],"date-time":"2016-02-09T10:37:18Z","timestamp":1455014238000},"page":"1343-1373","source":"Crossref","is-referenced-by-count":23,"title":["Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods"],"prefix":"10.1090","volume":"86","author":[{"given":"Carlo","family":"Garoni","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Carla","family":"Manni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Stefano","family":"Serra-Capizzano","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Debora","family":"Sesana","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hendrik","family":"Speleers","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"14","published-online":{"date-parts":[[2016,8,3]]},"reference":[{"key":"1","series-title":"Pure and Applied Mathematics (Amsterdam)","isbn-type":"print","volume-title":"Sobolev spaces","volume":"140","author":"Adams, Robert A.","year":"2003","ISBN":"https:\/\/id.crossref.org\/isbn\/0120441438","edition":"2"},{"issue":"2","key":"2","doi-asserted-by":"publisher","first-page":"746","DOI":"10.1137\/05063533X","article-title":"On the asymptotic spectrum of finite element matrix sequences","volume":"45","author":"Beckermann, Bernhard","year":"2007","journal-title":"SIAM J. 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Speleers, Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis, Tech. Report TW650 (2014), Dept. Computer Science, KU Leuven. Available online at https:\/\/lirias.kuleuven.be\/handle\/ 123456789\/461954."},{"key":"7","doi-asserted-by":"publisher","first-page":"230","DOI":"10.1016\/j.cma.2014.06.001","article-title":"Robust and optimal multi-iterative techniques for IgA Galerkin linear systems","volume":"284","author":"Donatelli, Marco","year":"2015","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"key":"8","doi-asserted-by":"publisher","first-page":"1120","DOI":"10.1016\/j.cma.2014.11.036","article-title":"Robust and optimal multi-iterative techniques for IgA collocation linear systems","volume":"284","author":"Donatelli, Marco","year":"2015","journal-title":"Comput. Methods Appl. Mech. Engrg.","ISSN":"https:\/\/id.crossref.org\/issn\/0045-7825","issn-type":"print"},{"issue":"300","key":"9","doi-asserted-by":"publisher","first-page":"1639","DOI":"10.1090\/mcom\/3027","article-title":"Spectral analysis and spectral symbol of matrices in isogeometric collocation methods","volume":"85","author":"Donatelli, Marco","year":"2016","journal-title":"Math. Comp.","ISSN":"https:\/\/id.crossref.org\/issn\/0025-5718","issn-type":"print"},{"key":"10","unstructured":"C. Garoni, Structured matrices coming from PDE approximation theory: spectral analysis, spectral symbol and design of fast iterative solvers, Ph.D. Thesis in Mathematics of Computation, University of Insubria, Como, Italy (2015). Available online at http:\/\/hdl.handle.net\/ 10277\/568."},{"issue":"4","key":"11","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"},{"key":"12","unstructured":"C. Garoni and S. Serra-Capizzano, The theory of Generalized Locally Toeplitz sequences: A review, an extension, and a few representative applications, Tech. Report 2015-023 (2015), Dept. Information Technology, Uppsala University. Available online at http:\/\/www.it.uu.se\/ research\/publications\/reports\/2015-023\/. This is the preliminary version of: C. Garoni and S. 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