BEGIN:VCALENDAR VERSION:2.0 PRODID:researchseminars.org CALSCALE:GREGORIAN X-WR-CALNAME:researchseminars.org BEGIN:VEVENT SUMMARY:Carla Manni (University of Rome Tor Vergata) DTSTART:20210420T070000Z DTEND:20210420T080000Z DTSTAMP:20260423T052642Z UID:CMWebinar/7 DESCRIPTION:Title: Spectral analysis of matrices from isogeometric methods\nby Carla Ma nni (University of Rome Tor Vergata) as part of Australian Seminar on Comp utational Mathematics\n\n\nAbstract\nWhen discretizing a linear PDE by a l inear numerical method\, the computation of the numerical solution reduces \nto solving a linear system. The size of this system grows when we refine the discretization mesh.\nWe are then in the presence of a sequence of li near systems with increasing size.\nIt is usually observed in practice tha t the corresponding sequence of discretization matrices enjoys\n an asympt otic spectral distribution. Roughly speaking this means that there exists a function\, say f\,\nsuch that the eigenvalues of the considered sequence of matrices behave like a sampling of f over an\nequispaced grid on the d omain of f\, up to some outliers.\n\nIsogeometric analysis is a well-estab lished paradigm for the analysis of problems governed by PDEs.\nIt provide s a design-through-analysis connection by exploiting a common representati on model. This connection is achieved\nby using the functions adopted in CAD systems not only to describe the domain geometry\, but also to represe nt the numerical solution of\nthe differential problem.\nIn its original f ormulation IgA is based on (tensor-product) B-splines and their rational e xtension\, the so-called NURBS [2].\n\nIn this talk we review the main spe ctral properties of discretization matrices arising from isogeometric meth ods\, based on \nd-variate NURBS of given degrees and applied to general s econd-order\nelliptic differential problems defined on a d-dimensional dom ain [4\,5]\, \ndiscussing the differences and the similarities with the FE M case [6]. \nWe also discuss the relation between outliers and convergenc e to eigenfunctions of classical differential operators under k-refinement . \n\nThe provided spectral information can be exploited for designing i terative solvers [3] \nwith convergence speed independent of the fineness parameters and also substantially independent of the degrees of the used N URBS\, [1].\n\nThe talk is based on joint works with C. Garoni\, F. Pelosi \, E. Sande\, H. Speleers\, S. Serra-Capizzano.\n\nReferences\n\n[1] N. Co llier\, L. Dalcin\, D. Pardo\, V.M. Calo\nThe cost of continuity: Performa nce of iterative solvers on isogeometric finite elements\, \nSIAM Journal on Scientific Computing\, 35 A767-A784\, 2013\n \n[2] J.A. Cottrell\, T.J. R. Hughes\, Y. Bazilevs\,\nIsogeometric Analysis: Toward Integration of CA D and FEA\,\nJohn Wiley & Sons\, 2009.\n\n[3] M. Donatelli\, C. Garoni\, C. Manni\, S. Serra-Capizzano\, H. Speleers\, \nSymbol-based multigrid met hods for Galerkin B-spline isogeometric analysis\, SIAM Journal on Numeric al Analysis\, 55\, 31-62\, 2017.\n\n[4] C. Garoni\, C. Manni\, F. Pelosi\, S. Serra-Capizzano\, H. Speleers\, \nOn the spectrum of stiffness matrice s arising from isogeometric analysis\, Numerische Mathematik\, 127\, 751-7 99\, 2014.\n\n[5] C. Garoni\, C. Manni\, S. Serra-Capizzano\, H. Speleers\ , NURBS in isogeometric discretization methods: A spectral analysis\, \nN umerical Linear Algebra with Application}\, 2020\;27:e2318.\n\n[6] C. Garo ni\, H. Speleers\, S-E. Ekstrom\, A. Reali\, S. Serra-Capizzano\, T.J.R. Hughes\, \nSymbol-based analysis of finite element and isogeometric B-spli ne discretizations of eigenvalue problems: \nExposition and review\, Archi ves of Computational Methods in Engineering\, 26\, 1639-1690\, 2019.\n\n[ 7] E. Sande\, C. Manni\, H. Speleers:\nSharp error estimates for spline approximation: explicit constants\, n-widths\, and eigenfunction convergen ce\,\nMathematical Models and Methods in Applied Sciences}\, 29\, 1175--1 205\, 2019\n LOCATION:https://researchseminars.org/talk/CMWebinar/7/ END:VEVENT END:VCALENDAR