Abstract
A multigrid method for linear systems stemming from the Galerkin B-spline discretization of classical second-order elliptic problems is considered. The spectral features of the involved stiffness matrices, as the fineness parameter h tends to zero, have been deeply studied in previous works, with particular attention to the dependencies of the spectrum on the degree p of the B-splines used in the discretization process. Here, by exploiting this information in connection with \(\tau \)-matrices, we describe a multigrid strategy and we prove that the corresponding two-grid iterations have a convergence rate independent of h for \(p=1,2,3\). For larger p, the proof may be obtained through algebraic manipulations. Unfortunately, as confirmed by the numerical experiments, the dependence on p is bad and hence other techniques have to be considered for large p.
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Notes
Take into account that these statements hold when the discretization steps \(h_i=1/n_i\) in each direction \(x_i\) tend to 0 with the same speed. This happens, for instance, when \(n_i=\nu _in,\ \varvec{\nu }:=(\nu _1,\ldots ,\nu _d)\) is fixed and \(n\rightarrow \infty \).
The first property holds because \(q_d\) is nonnegative and, by a direct computation,
$$\begin{aligned} \sum \nolimits _{{\widehat{{\varvec{\theta }}}}\in {\mathcal {M}}({\varvec{\theta }})\cup \{{\varvec{\theta }}\}}q_d({\widehat{{\varvec{\theta }}}})=2^d>0,\quad \forall {\varvec{\theta }}\in [0,\pi ]^d. \end{aligned}$$
References
Aricò, A., Donatelli, M.: A V-cycle multigrid for multilevel matrix algebras: proof of optimality. Numer. Math. 105, 511–547 (2007)
Aricò, A., Donatelli, M., Serra-Capizzano, S.: V-cycle optimal convergence for certain (multilevel) structured linear systems. SIAM J. Matrix Anal. Appl. 26, 186–214 (2004)
Axelsson, O., Lindskog, G.: On the rate of convergence of the preconditioned conjugate gradient method. Numer. Math. 48, 499–523 (1986)
Beckermann, B., Serra-Capizzano, S.: On the asymptotic spectrum of finite element matrix sequences. SIAM J. Numer. Anal. 45, 746–769 (2007)
Bhatia, R.: Matrix Analysis. Springer, New York (1997)
Boor, C. de: A Practical Guide to Splines. Springer, New York (2001)
Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009)
Donatelli, M.: An algebraic generalization of local Fourier analysis for grid transfer operators in multigrid based on Toeplitz matrices. Numer. Linear Algebra Appl. 17, 179–197 (2010)
Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Robust and optimal multi-iterative techniques for IgA Galerkin linear systems. Comput. Methods Appl. Mech. Eng. 284, 230–264 (2015)
Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Robust and optimal multi-iterative techniques for IgA collocation linear systems. Comput. Methods Appl. Mech. Eng. 284, 1120–1146 (2015)
Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis, submitted
Fiorentino, G., Serra, S.: Multigrid methods for Toeplitz matrices. Calcolo 28, 283–305 (1991)
Fiorentino, G., Serra, S.: Multigrid methods for symmetric positive definite block Toeplitz matrices with nonnegative generating functions. SIAM J. Sci. Comput. 17, 1068–1081 (1996)
Gahalaut, K.P.S., Kraus, J.K., Tomar, S.K.: Multigrid methods for isogeometric discretization. Comput. Methods Appl. Mech. Eng. 253, 413–425 (2013)
Garoni, C.: 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. http://hdl.handle.net/10277/568 (2015)
Garoni, C., Manni, C., Pelosi, F., Serra-Capizzano, S., Speleers, H.: On the spectrum of stiffness matrices arising from isogeometric analysis. Numer. Math. 127, 751–799 (2014)
Garoni, C., Manni, C., Serra-Capizzano, S., Sesana, D., Speleers, H.: Spectral analysis and spectral symbol of matrices in isogeometric Galerkin methods. Technical Report 2015-005, Department of Information Technology, Uppsala University, Sweden (2015)
Hughes, T.J.R., Cottrell, J.A., Bazilevs, Y.: Isogeometric analysis: CAD, finite elements, NURBS, exact geometry and mesh refinement. Comput. Methods Appl. Mech. Eng. 194, 4135–4195 (2005)
Jin, X.Q.: Developments and Applications of Block Toeplitz Iterative Solvers. Kluwer Academic Publishers, Dordrecht (2002)
Ruge, J.W., Stüben, K.: Algebraic multigrid, Chapter 4 of the book Multigrid Methods by S. McCormick. SIAM Publications, Philadelphia (1987)
Serra, S.: Multi-iterative methods. Comput. Math. Appl. 26, 65–87 (1993)
Serra-Capizzano, S.: Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences. Numer. Math. 92, 433–465 (2002)
Serra-Capizzano, S.: Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 366, 371–402 (2003)
Serra-Capizzano, S.: The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl. 419, 180–233 (2006)
Serra-Capizzano, S., Tablino-Possio, C.: Spectral and structural analysis of high precision finite difference matrices for elliptic operators. Linear Algebra Appl. 293, 85–131 (1999)
Serra-Capizzano, S., Tablino-Possio, C.: Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs. Contemp. Math. 281, 295–318 (2001)
Acknowledgments
This work was partially supported by INdAM-GNCS Gruppo Nazionale per il Calcolo Scientifico, by the MIUR ‘Futuro in Ricerca 2013’ Programme through the project DREAMS, by the MIUR-PRIN 2012 N. 2012MTE38N, and by the Donation KAW 2013.0341 from the Knut & Alice Wallenberg Foundation in collaboration with the Royal Swedish Academy of Sciences, supporting Swedish research in mathematics.
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Communicated by Artem Napov, Yvan Notay, and Stefan Vandewalle.
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Donatelli, M., Garoni, C., Manni, C. et al. Two-grid optimality for Galerkin linear systems based on B-splines. Comput. Visual Sci. 17, 119–133 (2015). https://doi.org/10.1007/s00791-015-0253-z
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DOI: https://doi.org/10.1007/s00791-015-0253-z