Two-grid optimality for Galerkin linear systems based on B-splines. (English) Zbl 1388.65176
Summary: 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 \(p\).
MSC:
| 65N55 | Multigrid methods; domain decomposition for boundary value problems involving PDEs |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
| 65F08 | Preconditioners for iterative methods |
References:
| [1] | Aricò, A., Donatelli, M.: A V-cycle multigrid for multilevel matrix algebras: proof of optimality. Numer. Math. 105, 511-547 (2007) · Zbl 1114.65033 · doi:10.1007/s00211-006-0049-7 |
| [2] | 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) · Zbl 1105.65322 · doi:10.1137/S0895479803421987 |
| [3] | Axelsson, O., Lindskog, G.: On the rate of convergence of the preconditioned conjugate gradient method. Numer. Math. 48, 499-523 (1986) · Zbl 0564.65017 · doi:10.1007/BF01389448 |
| [4] | Beckermann, B., Serra-Capizzano, S.: On the asymptotic spectrum of finite element matrix sequences. SIAM J. Numer. Anal. 45, 746-769 (2007) · Zbl 1140.65080 · doi:10.1137/05063533X |
| [5] | Bhatia, R.: Matrix Analysis. Springer, New York (1997) · Zbl 0863.15001 · doi:10.1007/978-1-4612-0653-8 |
| [6] | Boor, C. de: A Practical Guide to Splines. Springer, New York (2001) · Zbl 0987.65015 |
| [7] | Cottrell, J.A., Hughes, T.J.R., Bazilevs, Y.: Isogeometric Analysis: Toward Integration of CAD and FEA. Wiley, Chichester (2009) · Zbl 1378.65009 |
| [8] | 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) · Zbl 1240.65353 |
| [9] | 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) · Zbl 1425.65136 · doi:10.1016/j.cma.2014.06.001 |
| [10] | 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) · Zbl 1425.65196 · doi:10.1016/j.cma.2014.11.036 |
| [11] | Donatelli, M., Garoni, C., Manni, C., Serra-Capizzano, S., Speleers, H.: Symbol-based multigrid methods for Galerkin B-spline isogeometric analysis, submitted · Zbl 1355.65153 |
| [12] | Fiorentino, G., Serra, S.: Multigrid methods for Toeplitz matrices. Calcolo 28, 283-305 (1991) · Zbl 0778.65021 · doi:10.1007/BF02575816 |
| [13] | 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) · Zbl 0858.65039 · doi:10.1137/S1064827594271512 |
| [14] | Gahalaut, K.P.S., Kraus, J.K., Tomar, S.K.: Multigrid methods for isogeometric discretization. Comput. Methods Appl. Mech. Eng. 253, 413-425 (2013) · Zbl 1297.65153 · doi:10.1016/j.cma.2012.08.015 |
| [15] | 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) · Zbl 1297.65153 |
| [16] | 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) · Zbl 1298.65172 · doi:10.1007/s00211-013-0600-2 |
| [17] | 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) · Zbl 1359.65252 |
| [18] | 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) · Zbl 1151.74419 · doi:10.1016/j.cma.2004.10.008 |
| [19] | Jin, X.Q.: Developments and Applications of Block Toeplitz Iterative Solvers. Kluwer Academic Publishers, Dordrecht (2002) |
| [20] | Ruge, J.W., Stüben, K.: Algebraic multigrid, Chapter 4 of the book Multigrid Methods by S. McCormick. SIAM Publications, Philadelphia (1987) |
| [21] | Serra, S.: Multi-iterative methods. Comput. Math. Appl. 26, 65-87 (1993) · Zbl 0790.65025 · doi:10.1016/0898-1221(93)90035-T |
| [22] | Serra-Capizzano, S.: Convergence analysis of two-grid methods for elliptic Toeplitz and PDEs matrix-sequences. Numer. Math. 92, 433-465 (2002) · Zbl 1013.65026 · doi:10.1007/s002110100331 |
| [23] | Serra-Capizzano, S.: Generalized locally Toeplitz sequences: spectral analysis and applications to discretized partial differential equations. Linear Algebra Appl. 366, 371-402 (2003) · Zbl 1028.65109 · doi:10.1016/S0024-3795(02)00504-9 |
| [24] | Serra-Capizzano, S.: The GLT class as a generalized Fourier analysis and applications. Linear Algebra Appl. 419, 180-233 (2006) · Zbl 1109.65032 · doi:10.1016/j.laa.2006.04.012 |
| [25] | 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) · Zbl 0940.65113 · doi:10.1016/S0024-3795(99)00022-1 |
| [26] | Serra-Capizzano, S., Tablino-Possio, C.: Positive representation formulas for finite difference discretizations of (elliptic) second order PDEs. Contemp. Math. 281, 295-318 (2001) · Zbl 0994.65112 · doi:10.1090/conm/281/04664 |
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